MAXIMUM NORM A POSTERIORI ERROR ESTIMATE FOR A 3D SINGULARLY PERTURBED SEMILINEAR REACTION-DIFFUSION PROBLEM
|
|
- Paula Walker
- 5 years ago
- Views:
Transcription
1 MAXIMUM NORM A POSTERIORI ERROR ESTIMATE FOR A 3D SINGULARLY PERTURBED SEMILINEAR REACTION-DIFFUSION PROBLEM NARESH M. CHADHA AND NATALIA KOPTEVA Abstract. A singularly perturbed semilinear reaction-diffusion problem in the unit cube, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio condition is imposed. This result is obtained by combining (i) sharp bounds on the Green s function of the continuous differential operator in the Sobolev W, and W, norms and (ii) a special representation of the residual in terms of an arbitrary current mesh and the current computed solution. Numerical results on a priori chosen meshes are presented that support our theoretical estimate. Key words. Semilinear reaction-diffusion, singular perturbation, a posteriori error estimate, maximum norm, no mesh aspect ratio condition, finite differences, layer-adapted mesh. AMS subject classifications. 65N06, 65N5, 65N50. Introduction We focus on the following singularly perturbed semilinear reaction-diffusion problem posed in the unit cube: (.) T u := ε u + b(x, u) = 0, x = (x, x, x 3 ) Ω = (0, ) 3, u(x) = 0, x Ω. Here ε is a small positive parameter, = /x + /x + /x 3 is the standard Laplace operator, the function b is sufficiently smooth and satisfies (.) 0 < β < b u (x, u) β for all (x, u) [0, ] 3 R. Under condition (.), problem (.) has a unique solution, which exhibits sharp boundary layers of width O(ε ln ε ) along the boundary Ω. The aim of the present paper is to extend the two-dimensional a posteriori error estimate of [9] to three dimensions. This result is obtained by combining a special representation of the residual and sharp bounds on the Green s function of the linearized continuous differential operator in the Sobolev W, and W, norms. Compared to [9], the main difficulties in the present paper lie in the analysis of the Green s function. First, we use the explicit fundamental solution for the constantcoefficient operator ε + γ, and it is different in two and three dimensions. Furthermore, some parts of the analysis [9] for the variable-coefficient case, e.g., [9, 3.], do not yield the desired estimates in three dimensions (one gets additional negative powers of ε in the right-hand sides of the asserted bounds). Other parts This publication has emanated from research conducted with the financial support of Science Foundation Ireland under the Research Frontiers Programme 008; Grant 08/RFP/MTH536. Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland, naresh.chadha@ul.ie, natalia.kopteva@ul.ie.
2 of [9] were in some ways simplified in the current paper. We refer the reader to Remark 3. for a more detailed comparison. In a more general context, we note that sharp estimates for continuous Green s functions (or their generalized versions) frequently play a crucial role in a priori and a posteriori error analyses [, 4, 7]. Our error estimate will be in the maximum norm, which is sufficiently strong to capture layers and hence seems most appropriate for layer solutions. (The few known a posteriori error estimates for anisotropic meshes are in a weaker energy norm; see, e.g., [, ].) We also refer the reader to related papers on maximum norm a posteriori error estimation in one dimension [6, 7, 0, 4]. Another essential feature of our error estimate, is that we assume no mesh aspect ratio condition. It is crucial in the context of layer solutions, as for problems of type (.), a priori error analyses [6, 5, 8, 8] show that ε-uniform numerical approximations of layer solutions can be obtained using relatively small ε-independent numbers of mesh nodes. But this is attained by using a priori meshes that are anisotropic in layer regions, i.e. include extremely thin mesh cells that have extremely high aspect ratios (typically O(ε )). Note that a posteriori error estimates are typically obtained under the shape-regularity condition (equivalent to bounded-mesh-aspect-ratio condition) [, 3,, 7], so such estimates do not seem very suitable for constructing efficient layer-adapted meshes. We make no attempt to suggest or analyze any particular adaptive mesh generation algorithm. But we note that many successful algorithms are based on interpolation error estimates such as presented in [0, 9, 7], roughly speaking, the criterion on the generated mesh being a small interpolation error. Thus the generated, possibly, anisotropic mesh is supposed to be (quasi-)uniform under the metric induced by the positive definite Hessian matrix of the solution (or its scaled majorant); see, e.g., [6,, 5, 9]. It should be noted that such algorithms are not completely theoretically justified. E.g., the relation of the actual error of a numerical method to the interpolation error under no mesh aspect ratio condition is still to be established for many problems, in particular, in the maximum norm. Furthermore, linear interpolation error bounds involve the Hessian matrix of the unknown exact solution, which is replaced in the adaptive algorithm by its computed-solution analogue. To theoretically justify this replacement, one still needs to establish Hessian-matrix recovery formulas under no mesh aspect ratio condition, which are not available in the literature. An alternative theoretical justification, to which this paper aims to contribute, might be given by a posteriori error estimates that hold true under no mesh aspect ratio condition and directly relate the actual error to a certain discrete linearinterpolation-error-bound analogue, which involves the local mesh sizes and certain computed-solution approximations of the second-order derivatives. Indeed, roughly speaking, our a posteriori error estimate (.3), (.4) below is of this type, i.e. might be viewed as a discrete analogue of the linear interpolation error estimates. Our problem (.) will be discretized using the standard second-order seven-point difference scheme (see (.) below) on an arbitrary tensor-product mesh x ij l } in ) with 0 = x[0] s < x [] s < < x [N s] = for [0, ] 3, where x ij l = (x [i], x[j], x[l] 3 s =,, 3, while h i = x [i] x[i ], τ j = x [j] x[j ] and k l = x [l] 3 x[l ] 3 are the local mesh sizes. Note that such tensor-product meshes present an idealized situation, as practical a posteriori mesh construction algorithms use either irregular meshes or curvilinear s
3 3 tensor-product meshes. Therefore we consider our error estimate more interesting from a theoretical point of view since it shows that the bounded-mesh-aspect-ratio condition is not essential for a posteriori errors (as well as for interpolation errors). Also, if (possibly-curvilinear) tensor-product meshes are used in layer regions, where the mesh adaptation is most needed, one might conjecture that a local version of our a posteriori error estimate would apply there. Our main result is the following maximum norm a posteriori error estimate, in which the error is understood as the difference between the exact solution and the trilinear interpolant of the computed solution: τ (.3) U I u C 0 [ max i=,...,n j=0,...,n l=0,...,n 3 h i M,ij l } + max see Theorem.. Here, roughly speaking, i=0,...,n j=,...,n l=0,...,n 3 j M,ij l } + max i=0,...,n j=0,...,n l=,...,n 3 (.4) M s,ij l D su ij l ln ( + ε/κ ) +, s =,, 3, k l M 3,ij l } ] ; with κ = minminh i }, minτ j }, mink l }}. By U I we denote the trilinear interpolant of the computed solution U (the finite difference computed solution is orig- i j l inally defined at the mesh nodes only; hence to measure the error in the entire domain, one first has to interpolate the computed solution there). The quantities DsU ij l for s =,, 3 are the standard discrete approximations of u/xs defined in (.3). In (.4), a few terms are skipped, for which the one-dimensional analysis [7] and the numerical results of [9] and 6 suggest that they are less important; see Theorem. for the precise definitions of M s,ij l. The error constant C 0 in (.3) is independent of ε, the mesh and aspect ratios of its elements, but we do not specify its value. (Note that for singularly perturbed problems, the error constant may grow as ε becomes small; hence that it is ε-uniform is more important than its precise value.) The paper is organized as follows. In, we describe the numerical method, present our a posteriori error estimate in Theorem., and outline its proof. Next, in 3, we establish some sharp bounds on the Green s function of a linearized version of (.) in the Sobolev W, and W, norms. They imply certain stability properties of the differential operator T from (.), which are presented in 4. Then in 5, we obtain a special representation of the residual in terms of an arbitrary current mesh and the current computed solution, and therefore complete the proof of Theorem.. Finally, in 6, numerical results on a priori chosen meshes are given that support our theoretical estimate. Notation. Let p ; Ω, where p, denote the standard L p ( Ω) norm of scalar or vector functions defined in any domain Ω R 3. Furthermore, the standard notation W k,p ( Ω) is used for the Sobolev spaces with the norm k,p ; Ω defined, for any scalar function v(x) in a domain Ω, by v k,p ; Ω = v p ; Ω + k l= v l,p ; Ω, k =,, v,p ; Ω = 3 s= x s v p ; Ω, v,p ; Ω = 3 s,t= x s x t v p ; Ω ; see, e.g., [3]. We shall use the notation p and k,p for p ;Ω and k,p ;Ω when there is no ambiguity. Sometimes the domain of interest will be an open ball B(a ; ρ) = B(a, a, a 3 ; ρ) = x : 3 s= (x s a s ) < ρ } centered at a of radius ρ.
4 4 Throughout the paper we let C denote a generic positive constant that may take different values in different formulas, but is always independent of the mesh and ε. A subscripted C (e.g., C ) denotes a positive constant that is independent of the mesh and ε and takes a fixed value. For any two quantities w and w, the notation w = O(w ) means w Cw. Remark.. The assumption b u (x, u) β in (.) can be omitted as it follows, for some constant β, from 0 < β < b u (x, u) and u being a unique and bounded solution of (.), (to be more precise, u β b(, 0) ); see, e.g., [30, ]. Note that assumption (.) enables us to linearize (.) and then invoke the Green s function in our analysis.. Numerical method. Main result We consider problem (.) under the standard compatibility conditions at the corners of the domain Ω: (.) b(x, 0) = 0 for x = (x, x, x 3 ) : x, x, x 3 0, }, which guarantee that u C 3 ( Ω). Numerical method. The computed solution U is required to satisfy the standard seven-point finite difference discretization of (.) (.) ε [D U ij l + D U ij l + D 3U ij l ] + b(x ij l, U ij l ) = 0, where i =,... N, j =,..., N, l =,..., N 3, subject to U ij l = 0 on the boundary, i.e. if i = 0, N or j = 0, N or l = 0, N 3. Here, as usual, U ij l is associated with the mesh node x ij l = (x [i], x[j], x[l] 3 ), and we use the standard finite difference operators, defined for any discrete function V ij l by (.3) D V ij l = V ij l V i,j,l, D h V ij l = D V i+,j,l D V ij l, i (h i + h i+ )/ D V ij l = V ij l V i,j,l, D τ V ij l = D V i,j+,l D V ij l, j (τ j + τ j+ )/ D3 V ij l = V ij l V i,j,l, D k 3V ij l = D 3 V i,j,l+ D3 V ij l. l (k l + k l+ )/ By (.), there exists a unique solution of the discrete problem (.) on an arbitrary mesh x ij l }; see, e.g., [5]. At this stage, D U ij l is defined for i =,..., N, j = 0,..., N, l = 0,..., N 3. Similarly D U ij l and D 3U ij l are defined for j =,..., N, i, l, and l =,..., N 3, i, j, respectively. We now extend D U ij l to the mesh nodes i = 0, N as follows. The zero boundary conditions imply that D su 0,j,l = D su N,j,l = 0 for s =, 3. In view of this, the discrete equation (.) formally extended to i = 0 and i = N, becomes (.4a) D U ij l := ε b(x ij l, 0) for i = 0, N, j = 0,..., N, l = 0,..., N 3. Similarly, we extend D U ij l to j = 0, N and D 3U ij l to l = 0, N 3 by (.4b) (.4c) D U ij l := ε b(x ij l, 0) for j = 0, N, i = 0,..., N, l = 0,..., N 3. D 3U ij l := ε b(x ij l, 0) for l = 0, N 3, i = 0,..., N, j = 0,..., N. Note that, by (.), the above relations (.4) imply that D U ij l = D U ij l = D 3U ij l = 0 at the corners of our domain, which is consistent with the boundary condition in (.).
5 5 Remark.. Now that DsU ij l, where s =,, 3, are extended by (.4) to all i, j, l, our discrete equation (.) holds true for all i = 0,..., N, j = 0,..., N and l = 0,..., N 3. Trilinear interpolation notation. Let U I = U I (x) be the standard trilinear interpolant of the computed solution U ij l, i.e. U I is continuous in Ω, trilinear on each (x [i ], x [i] ) (x[j ], x [j] ) (x[l ] 3, x [l] 3 ), and equal to U ij l at the mesh nodes: (.5) U I (x ij l ) = U ij l for i = 0,..., N, j = 0,..., N, l = 0,..., N 3. Similarly, we define the trilinear interpolant v I (x) for any discrete function v ij l or any continuous function v(x). Furthermore, we shall use the standard one-dimensional linear interpolants v I s with respect to x s for s =,, 3, that are defined, for any function v, as follows. For each fixed x, x 3 in the domain of v, we have v I (x [i], x, x 3 ) = v(x [i], x, x 3 ), and v I (x) is linear on each (x [i ], x [i] ). Similarly, vi (x, x [j], x 3) = v(x, x [j], x 3), v I 3 (x, x, x [l] 3 ) = v(x, x, x [l] 3 ) and, furthermore, vi and v I 3 are linear on each (x [j ], x [j] ) and (x[l ] 3, x [l] 3 ), respectively. Note that the trilinear interpolation can be represented as a product of the three one-dimensional interpolation operators independently of the order of the interpolation steps. In particular, for the trilinear interpolant U I of U ij l we have (.6) U I (x) = [ U I ] II3 = [ U I ] II3 = [ U I3 ] II. The next theorem gives a maximum norm a posteriori error estimate, which is the main result of the present paper. Note that this is an extension to three dimensions of [9, Theorem.]. Theorem.. Let u(x) be a solution of problem (.), (.), (.), U ij l a solution of discrete problem (.) on an arbitrary mesh x ij l }, and U I (x) its trilinear interpolant (.5). Then [ U I } u C 0 max h i=,...,n i M,ij l + max j=0,...,n l=0,...,n 3 where i=0,...,n j=,...,n l=0,...,n 3 } τ j M,ij l + } ] max k i=0,...,n l M 3,ij l, j=0,...,n l=,...,n 3 M,ij l := min D U i,j,l, D U ij l } ln ( + ε/κ ) + ε D D U ij l + D U ij l +, M,ij l := min D U i,j,l, D U ij l } ln ( + ε/κ ) + ε D D U ij l + D U ij l +, M 3,ij l := min D 3U i,j,l, D 3U ij l } ln ( + ε/κ ) + ε D 3 D 3U ij l + D 3 U ij l +, with κ :=minminh i }, minτ j }, mink l }}, and the constant C 0 is independent of ε i j l and the mesh. Proof outline. To simplify the presentation, here and throughout 4-5, in which this proof is continued, we assume that N = N = N 3 = N. In view of (.), one gets T U I T u = ε [ /x + /x + /x 3] U I + b(x, U I ). Note that here U I /x s, for s =,, 3, are understood in the sense of distributions. Next, we introduce an auxiliary function q(x) := b(x, U I (x))
6 6 and its trilinear interpolant q I on the mesh (x ij l )}. Now one has T U I T u = ε [ /x + /x + /x 3] U I + q I + [ q q I]. As q ij l := q(x ij l ) = b(x ij l, U ij l ), in view of Remark., the discrete equation (.) yields q ij l = ε [D U ij l + D U ij l + D 3U ij l ] for i, j, l = 0,..., N. This observation leads to the decomposition q ij l = q,ij l + q,ij l + q 3,ij l, where (.7) q s,ij l := ε D s U ij l for s =,, 3, i,j, l = 0,..., N, and then to q I (x) = q I (x) + q I (x) + q I 3(x) = [ q I ]I I 3 + [ q I ]I I 3 + [ q I3 3 ]I I, x Ω. Here we also used a version of (.6) for functions q s with s =,, 3. We now get T U I T u = [ ε U I +q I ] I I x 3+ [ ε U I +q I ] I I x 3+ [ ε U I 3 +q I ] I I x 3 + [ 3 3 q q I ]. To obtain this equation, we also used the following relations: (.8) U I = [ U I ] I x x I 3, U I = [ U I ] I x x I 3, U I = [ U I ] I x 3 x 3 I. 3 The above three relations follow from (.6) as any operator / x s is commutative with I t for t s (but not with I s ); see also Remark.3. The proof will be completed in 5 by representing the residual T U I T u as (.9) T U I T u = x F (x) + x F (x) + x 3 F 3 (x) + [ q q I], where F, F and F 3 are some functions of the current mesh and computed solution. In view of (.9), the error U I u will be estimated in the maximum norm by linearizing the operator T and invoking its stability properties. Stability of T will be addressed in 4 using certain bounds for the Green s function of 3. Remark.3. We understand U I /xs, for s =,, 3, in the sense of distributions. To be more precise, in (.8) we use U I = N [ ] x i= i DU ij l δ(x x [i] ), [ ] II U I 3 N = i= [ ] i D II 3 U ij l δ(x x [i] ), x where i := (h i + h i+ )/ and δ( ) is the Dirac δ-distribution. 3. Green s function To investigate the error U I u, we shall linearize (.9) and then invoke certain estimates of the Green s function of the resulting linear equation. These estimates are the main result of this section. Thus, throughout the section, we focus on a linear case of (.), where we set b(x, u) := p(x)u f(x) and thus arrive at (3.) Lu := ε u + p(x)u = f(x) in Ω, u = 0 on Ω. Here p L (Ω), and, in agreement with (.), it also satisfies (3.) 0 < β p(x) β. Let G(x; ξ) be the Green s function of the linear self-adjoint operator L. For each x = (x, x, x 3 ) Ω, it satisfies (3.3) L ξ G = ε ξ G + p(ξ)g = δ(x ξ), ξ Ω, G(x; ξ) = 0, ξ Ω,
7 7 where ξ = (ξ, ξ, ξ 3 ) and ξ = /ξ + /ξ + /ξ, while δ( ) is the threedimensional Dirac δ-distribution. We now have an explicit formula for the unique solution u of problem (3.): (3.4) u(x) = G(x; ξ) f(ξ) dξ, where we used the notation dξ = dξ dξ dξ 3. Theorem 3.. The Green s function G(x; ξ) from (3.3) satisfies (3.5a) G(x; ) ;Ω + ε G(x; ), ;Ω C. Ω Furthermore, for any ball B( x ; ρ) of radius ρ centered at any x Ω we have (3.5b) G(x; ), ;B( x ;ρ) Ω C ε ρ; while for the ball B(x ; ρ) of radius ρ centered at x, we have (3.5c) G(x; ), ;Ω\B(x ;ρ) C ε ln( + ε/ρ). This entire section is devoted to the proof of this theorem, which is the main result of the section. The proof is in two steps. First, in 3. we shall estimate the auxiliary Green s function Ḡ in the constant-coefficient case in the positive octant space using the explicit fundamental solution. The remaining part of the proof ( ) will, roughly speaking, deal with G Ḡ. Remark 3.. Note that the statement of Theorem 3. is precisely as in the twodimensional case [9, Theorem 3.]. The proof, however, is different in a few ways. First, we note that the fundamental solutions for the constant-coefficient operator ε + γ, which are used in both analyses, are different in two and three dimensions. Furthermore, the argument of [9, 3.] for a variable-coefficient case does not yield the desired estimates in three dimensions (it leads to additional negative powers of ε in the right-hand sides of our asserted bounds). For this case, we therefore give a completely different proof and even weaken the assumption x i p C of [9, 3.] to ε x i p C in 3.. The final part of the proof for the most general case of p (see 3.3) has evolved from [9, 3.3], and is simpler in the sense that now we avoid using any cut-off functions and deal either with G Ḡ or sometimes directly with G. 3.. Constant-coefficient case. The first step in the proof of Theorem 3. consists in establishing it for a particular constant-coefficient case. Set p := γ for some γ = const > 0, and let Ω be the positive octant space R 3 + = x, x, x 3 > 0}. We denote the differential operator in this particular case by L, and the Green s function by Ḡ. Thus for each x we have (3.6) Lξ Ḡ(x; ξ) := ε ξ Ḡ + γ Ḡ = δ(x ξ), ξ, ξ, ξ 3 > 0. The fundamental solution for the operator ξ + ν in R 3 is e νr /(4πr); see, e.g., [8, 8.3]. This readily provides the fundamental solution for our differential operator L, which is (3.7) g(x; ξ) := e γr/ε 4πε, r := (ξ x ) r + (ξ x ) + (ξ 3 x 3 ).
8 8 Now the Green s function for L over the octant is easily obtained by the method of images and involves eight terms of the type ±g(±x, ±x, ±x 3 ; ξ); to be more precise we have (3.8) Ḡ(x; ξ) = (σ σ σ 3 ) g ( x [σ,σ,σ 3 ] ; ξ ), x [σ,σ,σ 3 ] := (σ x, σ x, σ 3 x 3 ). σ,σ,σ 3=, Lemma 3.3. (i) For Ḡ(x; ξ) of (3.8), estimates (3.5) of Theorem 3. hold true, in which G is replaced by Ḡ. (ii) Furthermore, we have (3.9) Ḡ(x; ) ;Ω C ε 3/, Ḡ(x; ) ;Ω\B(x ;ρ) C ε 3/ e γρ/ε. Proof. It suffices to prove estimates (3.5) and (3.9) with G and Ḡ, respectively, replaced by the term g(x; ξ) of the representation (3.8) of Ḡ, as the estimates for the other seven terms are similar. Let the stretching transformation from ξ = (ξ, ξ, ξ 3 ) to the new coordinates ˆξ = (ˆξ, ˆξ, ˆξ 3 ) := (ξ x)/ε map any domain Ω R 3 into ˆΩ. Furthermore, consider a scaled version ĝ(ˆξ) of g(x; ξ) from (3.7) defined by (3.0) ĝ(ˆξ) := ε 3 g(x; ξ) = e γˆr, where ˆr := ˆξ 4π ˆr + ˆξ + ˆξ 3, so that g dξ = ĝ dˆξ, where dˆξ = dˆξ dˆξ dˆξ = ε 3 dξ. Therefore for any domain Ω we have (3.) g(x; ) k, ;Ω = ε k ĝ k, ; ˆΩ, g(x; ) ;Ω = ε 3/ ĝ ;ˆΩ. Now we shall establish parts (i) and (ii) of our lemma. (i) A calculation using the standard differentiation formulas ĝ ˆξ i = ĝ ˆr ˆr ˆξ i, ĝ ˆξ i ˆξ = ĝ j ˆr ˆr ˆξ ˆr i ˆξ + ĝ j ˆr ˆr ˆξ i ˆξ, j where ĝ ˆr = γˆr + 4π ˆr e γˆr ĝ, ˆr = γ ˆr + γˆr + 4π ˆr 3 and also ˆr/ ˆξ i = ˆξ i /ˆr and ˆr/( ˆξ i ˆξ j ) /ˆr, yields e γˆr ˆr ĝ C(γˆr + )e ˆξ γˆr, ˆr ĝ ˆr + C i ˆξ i ˆξ e γˆr. j ˆr Combining this with the first relation in (3.), we obtain the required analogues of (3.5) for g as follows. First, note that g(x; ) ;Ω + ε g(x; ), ;Ω = ĝ, ;ˆΩ C Similarly, we obtain 0 (γˆr + )e γˆr dˆr C. ρ/ε g(x; ), ;B( x ;ρ) g(x; ), ;B(x ;ρ) Cε (γˆr + )e γˆr dˆr Cε (ρ/ε); here replacing the integral over B( x ; ρ) by the integral over B(x ; ρ) yields an upper bound, since (γˆr + )e γˆr is a positive decreasing function. Finally, we get g(x; ), ;Ω\B(x ;ρ) Cε ρ/ε 0 ˆr + e γˆr dˆr ε ln( + ε/ρ). ˆr
9 9 (ii) A straightforward calculation using (3.0) shows that ĝ ; ˆΩ C e γˆr dˆr C, 0 ĝ ; ˆΩ\ ˆB(x C e γˆr dˆr Ce γρ/ε. ;ρ) Combining these with the second relation in (3.), we immediately get the required analogues of (3.9) for g. Remark 3.4. An inspection of the proof of Lemma 3.3, in which we used the explicit representation (3.8),(3.7) of the Green s function in the constant-coefficient case, shows that the estimates of the Green s function in Theorem 3. are sharp. 3.. Smooth-coefficient case. In this subsection, we shall use the estimates for the constant-coefficient Green s function Ḡ to establish a variable-coefficient case of Theorem 3. under the additional assumption that the coefficient p is differentiable and (3.) ε x i p C for i =,, 3. Lemma 3.5. If the coefficient p satisfies (3.) and (3.), then the Green s function G(x; ξ) from (3.3) satisfies estimates (3.5). Proof. Set γ := β in the definition (3.6) of Ḡ. Note that in addition to (3.3), G satisfies L x G(x; ξ) = ε x G+p(x)G = δ(x ξ) subject to G(x; ξ) = 0 for x Ω. Similarly, Ḡ satisfies L x Ḡ(x; ξ) = ε x Ḡ + γ Ḡ = δ(x ξ) subject to Ḡ(x; ξ) = 0 if x = 0 or x = 0 or x 3 = 0. Hence we have L x (Ḡ G) = [p γ ]Ḡ = [p β]ḡ 0, while Ḡ G = Ḡ 0 for x Ω. Now, applying the maximum principle, one gets Ḡ G 0, or 0 G Ḡ, so G(x; ) ;Ω Ḡ(x; ) ;Ω C, where the bound for Ḡ is given by Lemma 3.3. Next, let ξ [0, ]3 and construct a function (3.3) v(x; ξ) := G(x; ξ) ω(x) Ḡ(x; ξ), where ω(x) is a smooth cut-off function that equals for x [0, 3 4 ]3 and vanishes on the boundaries x =, x = and x 3 =. Thus v = 0 for x Ω. A calculation using L x G = L x Ḡ yields L x v = (γ p)ḡ + L x[( ω)ḡ], or where φ = φ + φ with ρ/ε L x v(x; ξ) = [ ε x + p(x)] v(x; ξ) = φ(x; ξ), φ (x; ξ) := [ γ p(x) ] Ḡ(x; ξ), φ (x; ξ) := L x [ ( ω(x)) Ḡ(x; ξ) ]. Comparing the problem for v with problem (3.) and recalling (3.4), we arrive at (3.4) v(x, ξ) = G(x; η) φ(η; ξ) dη. Applying ξ i, i =,, 3, to this formula, one gets (3.5) v(x; ), ;Ω G(x; ),Ω max η Ω φ(η; ), ;Ω. Ω We have already proved that G(x; ),Ω C. Note also that φ (η; ), ;Ω C Ḡ(η; ), ;Ω. Furthermore, if Ω [0, ]3 then φ (η; ), ;Ω C (the latter estimate holds as φ (η; ξ) = 0 for η [0, 3 4 ]3 ; otherwise, ξ η 4 so any derivative of φ is bounded by some C). Combining these observations with (3.5) and then
10 0 noting that (3.3) implies G(x; ), ;Ω at G(x; ), ;Ω v(x; ), ;Ω + Ḡ(x; ), ;Ω, we arrive C ( max η Ω Ḡ(η; ), ;Ω + ), where Ω [0, ]3. Setting Ω = [0, ]3 and Ω = B( x; ρ) [0, ]3 yields (3.5a), (3.5b) with Ω replaced by its subdomain [0, ]3. Dealing with the remaining seven cubic subdomains of Ω in a similar manner, one finally gets (3.5a), (3.5b). To establish the remaining estimate (3.5c), it suffices, by (3.3), to obtain v(x; ), ;Ω Cε. Let v = v + v, where v k, for k =,, is defined by (3.5) with v and φ respectively replaced by v k and φ k. Imitating our above argument, we get v (x; ), ;Ω C max η Ω φ (η; ), ;Ω C. We have, however, to modify our approach to estimate v (x; ), ;Ω. First, examining (3.7) and (3.8), we note that ξ g(±x, σ x, σ 3 x 3 ; ξ) = x g(±x, σ x, σ 3 x 3 ; ξ), ξ Ḡ(x; ξ) = x G(x; ξ), where G involve the same eight terms as Ḡ in (3.8), only with possibly different signs. Furthermore, G satisfies (3.5) (imitate the proof ξ g(x; ξ) = x g(x; ξ), so so of Lemma 3.3). In view of this, one gets ξ v (x, ξ) = G(x; η)γ p(η)} ξ Ḡ(η; ξ)dη = η [G(x; η)γ p(η)}] G(η; ξ)dη. Next, applying Ω ξ ξ i v (x; ) ;Ω ξ i, i =,, 3, to this formula yields Ω G(x; )γ p( )}, ;Ω max η Ω G(η; ), ;Ω Cε, where we used (3.), (3.) and (3.5a) that we already have for both G and G. As all second-order derivatives of v can be estimated in a similar manner, we get v (x; ), ;Ω Cε and thus complete the proof of (3.5c). We have now proved Theorem 3. under condition (3.). This condition is suitable for the particular linear case L of T. When we linearize (.9) in the general semilinear case, the coefficient p depends on u and U. It may satisfy (3.), but it is more convenient to avoid this assumption in our error analysis. Thus in the next subsection, we prove the general case of Theorem 3. under assumption (3.) only. Remark 3.6. One beneficial feature of the argument used in 3. is that it does not require a pointwise barrier for G (readily provided by Ḡ in 3.3). Instead, it suffices to have a sharp bound on G(x; ) ;Ω. This feature is significant, when the above argument is extended in a future paper to more complicated singularly perturbed convection-diffusion equations General case. Proof of Theorem 3.. Fix x = (x, x, x 3 ) Ω; without loss of generality, we shall consider only the case of x [0, /] 3, as the other cases are similar. Set γ := β in the definition (3.6) of Ḡ. Since, by Lemma 3.3(i), estimates (3.5) hold true for Ḡ, to get the desired estimates (3.5) for G, it suffices to show that (3.6a) ε (Ḡ G)(x; ), ;B(x ;ε) Ω + ε (Ḡ G)(x; ), ;B(x ;ε) Ω C, (3.6b) ε (Ḡ G)(x; ), ;B( x ;ρ) Ω C ρ/ε for ρ ε, (3.6c) ε G(x; ), ;Ω\B(x ;ε) + ε G(x; ), ;Ω\B(x ;ε) C.
11 Indeed, (3.5a) follows from its analogue for Ḡ combined with (3.6a) and (3.6c). The next estimate (3.5b) follows from (3.5a) if ρ > ε, and from its analogue for Ḡ combined with (3.6b) otherwise. Finally, estimate (3.5c) follows from (3.6c) for ρ ε, and is obtained combining its analogue for Ḡ with (3.6a) and (3.6c) otherwise. Now, to complete the proof, we shall establish each of the estimates in (3.6). (a) Note that, by (3.3) and (3.6), we get (3.7) L ξ (Ḡ G) = [ ε ξ + p(ξ) ] (Ḡ G) = [p(ξ) β] Ḡ. Therefore, by (3.), we have L ξ (Ḡ G) 0. Combining this with Ḡ G 0 on Ω and then applying the maximum/comparison principle yields 0 G Ḡ. Next, using the variable ˆξ = (ξ x)/ε and the notation ŵ(ˆξ) := w(ξ) for any function w, rewrite (3.7) in terms of the variable ˆξ as [ + ˆp ]( ˆḠ Ĝ) = [ˆp β] ˆḠ, or ( ˆḠ Ĝ) = ˆp ˆḠ βĝ. Now, an application of [3, Lemma 8. (Chap. 3, p. 8)] yields (3.8) ˆḠ Ĝ, ; ˆB(x ;ε) ˆΩ C [ ( ˆḠ Ĝ) ; ˆB(x ;ε) ˆΩ + ˆḠ ] Ĝ ; ˆB(x ;ε) ˆΩ, where the constant C is independent of ε since dist( ˆB(x ; ε), ˆB(x ; ε)) =. Note that the condition of [3, Lemma 8.] that ˆḠ Ĝ = 0 for ˆξ ˆΩ ˆB(x ; ε), or, equivalently, Ḡ G = 0 for ξ Ω B(x ; ε), is immediately satisfied for ε < /4 since x [0, /] 3 (otherwise, if ε [/4, ], one gets a version of (3.8) for ˆB(x ; ε ) and ˆB(x ; ε ) with ε := ε/5 < /4; then an obvious modification of our further argument will again yield (3.5)). To estimate the right-hand side in (3.8), recall that 0 G Ḡ, thus 0 Ĝ ˆḠ, so ˆḠ Ĝ ˆḠ and ( ˆḠ Ĝ) = ˆp ˆḠ βĝ C ˆḠ. These observations lead to (3.9) ˆḠ Ĝ, ; ˆB(x ;ε) ˆΩ C ˆḠ ; ˆB(x ;ε) ˆΩ. Rewriting this in terms of the original variable ξ, we get ε (Ḡ G)(x; ), ;B(x ;ε) Ω + ε (Ḡ G)(x; ), ;B(x ;ε) Ω C Ḡ(x; ) ;B(x ;ε) Ω, where Ḡ(x; ) ;B(x ;ε) Ω C ε 3/, by the first estimate in (3.9). Combining this with Ḡ G k, ;B(x ;ε) Ω Cε 3/ Ḡ G k, ;B(x ;ε) Ω, for k =,, yields (3.6a). (b) Let x be an arbitrary point in Ω and x [0, /] 3 (as the other cases are similar). To show (3.6b), imitate the argument used to prove (3.6a) with B(x ; ε) and B(x ; ε) replaced by B( x ; ρ) and B( x ; ρ + ε), invoking Ḡ G k, ;B( x ;ρ) Cρ 3/ Ḡ G k, ;B( x ;ρ) and ρ/ε. (c) Let ρ j := j ε and divide the domain Ω\B(x ; ε) into the non-overlapping subdomains D j := ξ Ω : ρ j < r < ρ j+ } where j = 0,,.... Furthermore, D j D j := D j D j D j+, so that dist(d j \Ω, D j\ω) ε/. Let the stretching transformation from ξ to ˆξ = (ξ x)/ε map D j into ˆD j. Rewriting the equation from (3.3) for ξ x in terms of the stretched variable ˆξ as Ĝ + ˆp Ĝ = 0 yields (3.0) Ĝ, ; ˆD j C Ĝ ; ˆD ; j
12 see [3, Lemma 8. (Chap. 3, p. 8)]; here the constant C is independent of ε since dist( ˆD j \ ˆΩ, ˆD j \ ˆΩ) /. Note that the condition of [3, Lemma 8.] that Ĝ = 0 on ˆD j ˆΩ is satisfied due to the boundary condition in (3.3). Rewritten in terms of the original variable ξ, estimate (3.0) implies that (3.) ε G(x; ), ;Dj + ε G(x; ), ;Dj C G(x; ) ;D j C Ḡ(x; ) ;D j. where we also used G Ḡ. Noting that D j Ω\B(x ; ρ j ) and recalling the second estimate in (3.9), we get Ḡ(x; ) ;D j Cε 3/ e γρj /ε. Combining this with (3.) and G k, ;Dj Cρ 3/ j G k, ;Dj, for k =,, we arrive at j=0 ε G(x; ), ;Dj + ε G(x; ), ;Dj Cρ 3/ j ε 3/ e γρ j /ε. Now, the required estimate (3.6c) is obtained recalling that Ω\B(x ; ε) = j=0 D j and noting that we have ( γρj ) 3/ e γρ γ(ρ j /ε j ρ j ) C e γρj/(4ε) C e s ds, ε 4ε j=0 since s / e s Ce s and ρ j = (ρ j ρ j ), and for the decreasing function e s the right Riemann sum gives a lower estimate of the corresponding integral. 4. Stability properties of differential operators In this section, we are concerned with subtle stability properties of the semilinear differential operator T from (.). The main result of this section, Theorem 4., will be applied in 5 to equation (.9), which relates the exact solution and the computed solution. Suppose the right-hand side f is of the special form (4.a) f(x) = [F (x) + F (x) ] + f(x), where F = (F, F, F 3 ) and F = ( F, F, F 3 ) are vector functions, whose components together with f are in L (Ω), and F = F /x + F /x + F 3 /x 3. Furthermore, we assume that (4.b) F (x) F 3 (x) F (x) = A i (x, x 3 ) (x x [i /] ) for x (x [i ], x [i ] ) [0, ] [0, ], (4.c) = B j (x, x 3 ) (x x [j /] ) for x [0, ] (x [j ], x [j ] ) [0, ], (4.d) = Q l (x, x ) (x 3 x [l /] 3 ) for x [0, ] [0, ] (x [l ] 3, x [l ] 3 ), where i, j, l =,..., N, respectively, and the notation x s [i /] is used with s =,, 3. γ/8 := (x [i ] s + x [i ] s )/ Theorem 4.. Suppose the function b in the definition (.) of T satisfies (.), and f is defined by (4.). Then, for any functions v, w W, (Ω) such that T v(x) T w(x) = f(x), and v = w on Ω, we have v w Cε [ E + E + E 3 ] ln ( + ε/κ ) + Cε F + β f, where κ = minminh i }, minτ j }, mink l }} and i j l E := max h i max A i(x, x 3 ) }, E := max i=,...,n x,x 3 [0,] E 3 := max l=,...,n j=,...,n k l max x,x [0,] Q l(x, x ) }. τ j max B j(x, x 3 ) }, x,x 3 [0,]
13 The above theorem is a three-dimensional version of [9, Theorem 4.]. The proof is in lines with the one in [9], so we only sketch it below for completeness. Proof. Using the standard linearization technique, one gets T v T w = L [ v w ], where the linear operator L is defined by (3.), in which, by (.), the coefficient p(x) satisfies (3.). As we now have the linear equation L[u v] = f, we shall deal with various components of f separately and, in particular, invoke the Green s function G of the operator L in our analysis. First we note that if F = (F, F, F 3 ) := 0 in (4.a), then (4.) u v Cε F + β f. This is easily shown by imitating the proof of [9, Lemma 4.], more specifically, by combining (3.4) with estimate (3.5a). Next, we claim that if F := 0 and f := 0 in (4.), then (4.3) u v Cε [ E + E + E 3 ] ln ( + ε/κ ). Combining this with the observation (4.) yields the assertion of the theorem. Thus it remains to prove (4.3). We get (4.3) by extending the proof from [9, Lemma 4.3] to three dimensions as briefly described below. Note that it suffices to get (4.3) only in the case of f := F /x, i.e. F = F 3 := 0, as the cases of f := F s /x s, for s =, 3, are similar. Fix x and denote v(ξ) := G(x; ξ). Then, using (3.4), one gets N (u v)(x) = F (ξ) v ξ (ξ) dξ = Ω i= dξ dξ 3 A i (ξ, ξ 3 ) For the integral in ξ, a calculation shows that (4.4) [i] x x [i ] (ξ x [i /] h ) v ξ (ξ) dξ i x [i] x [i ] x [i] x [i ] 3 (ξ x [i /] ) v ξ (ξ) dξ. v ξ ξ (ξ) dξ. However, we have to be careful when integrating v ξ ξ = G ξ ξ as this function has such a singularity at ξ = x that it is not in L (Ω). Thus we form a rectangularbox neighbourhood Ω of ξ = x (of diameter not exceeding O(κ)). Outside this neighbourhood the integral F v ξ dξ is estimated using (4.4) and then (3.5c). Over this neighbourhood, the integral F v ξ dξ is estimated using (3.5b). This completes the proof of (4.3) in the case of f := F /x. 5. Analysis of the numerical method. Proof of Theorem. To complete the proof of our main result, Theorem., which we started in, we shall invoke the following lemma. Lemma 5.. [7, 9] We have ε x U I s + q I s s s = x s F s, s =,, 3, where the semi-discrete functions F = F (x, x [j], x[l] 3 ), F = F (x [i], x, x [l] 3 ) and F 3 = F 3 (x [i], x[j], x 3) are defined by (5.a) F := q,ij l (x x [i /] ) + D q,ij l (x [i] x ), x (x [i ], x [i] ), for i =,..., N and j, l = 0,..., N,
14 4 (5.b) F := q,ij l (x x [j /] ) + D q,ij l (x [j] x ), x (x [j ], x [j] ), for j =,..., N and i, l = 0,..., N, (5.c) F 3 := q 3,ij l (x 3 x [l /] 3 ) + D 3 q 3,ij l (x [l] 3 x 3), x 3 (x [l ] 3, x [l] 3 ), for l =,..., N and i, j = 0,..., N. Proof. Imitate the proofs of [7, Theorem 3.3] and [9, Lemma 5.]. Remark 5.. One can easily check that F s, for s =,, 3, of (5.) allow an alternative representation: F = [q ] i,j,l (x x [i /] ) + [D q,ij l] O(h i ), x (x [i ], x [i] ), F = [q ] i,j,l (x x [j /] ) + [D q,ij l] O(τ j ), x (x [j ], x [j] ), F 3 = [q 3 ] i,j,l (x 3 x [l /] 3 ) + [D 3 q 3,ij l] O(k l ), x 3 (x [l ] 3, x [l] 3 ). Here, e.g., the new representation of F follows from q,ij l = [q ] i,j,l +h i [D q,ij l]. Proof of Theorem. (continued from ). Extend F s, s =,, 3, of Lemma 5. onto the whole domain Ω by the trilinear interpolation F (x) := [ F (x, x [j], x[l] 3 )] I I 3, F (x) := [ F (x [i], x, x [l] 3 )] I I 3, F 3 (x) := [ F 3 (x [i], x[j], x 3) ] I I. Now, noting that any operator /x s is commutative with I t for t s, we obtain the representation (.9) for the residual T U I T u. So, by Theorem 4., one gets (5.) U I u Cε [ E + E + E 3 ] ln ( + ε/κ ) + Cε Ē + β q q I, where (5.3a) E = max h i max i=,...,n x,x 3 [0,] (q,ij l) I I 3 } = max i=,...,n j,l=0,...,n h i q,ij l }, and similarly (5.3b) E = max τ j q,ij l }, E 3 = max k l q 3,ij l }, j=,...,n i,l=0,...,n l=,...,n i,j=0,...,n while Ē = max i=,...,n j,l=0,...,n h i D q,ij l } + max τ j=,...,n i,l=0,...,n j D q,ij l } + max l=,...,n i,j=0,...,n k l D 3 q 3,ij l }. Furthermore, in view of Remark 5. (compare it with (5.)), we observe that the quantities q s,ij l, s =,, 3, in (5.3) can be replaced by min q,i,j,l, q,ij l }, min q,i,j,l, q,ij l } and min q 3,i,j,l, q 3,ij l }, respectively. This yields a sharper version of (5.), (5.3), which is then combined with (.7). Now, to get the desired a posteriori error estimate of Theorem., it remains to show the trilinear interpolation estimate q q I C [ h i ( + D U ij l ) } + max j=,...,n i,l=0,...,n max i=,...,n j,l=0,...,n τ j ( + D U ij l ) } + max j=,...,n i,l=0,...,n k l ( + D 3 U ij l ) }]. This estimate follows from q q I = [q q I ] + [q I (q I ) I ] + [q II (q II ) I3 ] combined with the observation that q/x s C( + D s U ij l ) in each mesh
15 5 Table 6.. Bakhvalov mesh, λ = 3: maximum norm error e and the efficiency constant e/η for the upper error estimator η. ε = 0 ε = 0 ε = 0 3 ε = 0 k, k = 4,..., 0 N e e/η e e/η e e/η e e/η 6.74e-.80e-.7e-.3e-.73e-.3e-.73e-.3e e-3.8e- 4.3e-3.8e- 4.36e-3.8e- 4.38e-3.8e- 64.7e-3.86e-.08e-3.6e-.09e-3.6e-.0e-3.7e- 8.96e-4.87e-.70e-4.6e-.74e-4.6e-.75e-4.6e- Table 6.. Bakhvalov mesh, λ = 3: upper maximum norm error estimator η, its components η, η, η 3, and its efficiency constant e/η. ε = 0 ε = 0 k, k = 4,..., 0 N η η η 3 = η e/η η η η 3 = η e/η e- 7.79e- 9.64e-.80e-.0e-.07e-.33e-.3e- 3.9e-.4e-.48e-.8e- 3.0e- 3.09e- 3.4e-.8e e e-3 6.9e-3.86e- 8.6e-3 8.5e e-3.7e e-4.55e-3.58e-3.87e-.3e-3.3e-3.8e-3.6e- cell (x [i ], x [i] ) (x[j ], x [j] ) (x[l ] 3, x [l] 3 ); see [, Comment.5] for a similar argument. 6. Numerical results Our main result, the maximum norm a posteriori error estimate of Theorem., can be rewritten as (6.) e := U I u Cη, η := maxη 0, η, η, η 3 }, η n := max max h i=,...,n i M (n),ij l} ; max τ i=0,...,n j M (n),ij l} ; max k i=0,...,n l M (n) 3,ij l} }, j=0,...,n l=0,...,n 3 j=,...,n l=0,...,n 3 j=0,...,n l=,...,n 3 for n = 0,,, 3. Here we use C = C ln ( + ε/κ ), M (),ij l := min D U i,j,l, D U ij l }, M (),ij l := min D U i,j,l, D U ij l }, M () 3,ij l := min D 3U i,j,l, D 3U ij l }, and for s =,, 3, we also use M (0) () s,ij l =, M s,ij l := D s U ij l, M (3) s,ij l := ε D s DsU ij l. Note that the quantities η n involve M (n), n =,, 3, which can be viewed as discrete analogues of (possibly scaled) nth-order derivatives. We give numerical results on a priori chosen meshes to illustrate the efficiency of the upper maximum norm error estimator η in (6.) and its particular components η n, n = 0,,, 3. We are also interested in which of η n is the principal component in η if any. We shall compute the errors e and, more importantly, the quantities η, e/η, η n, e/η n and then examine their dependence on ε, numbers of mesh nodes and particular meshes.
16 6 Table 6.3. Uniform mesh: maximum norm error e and the efficiency constant e/η for the component η of the upper maximum norm error estimator η. ε = 0 ε = 0 3 ε = 0 4 ε = 0 k, k = 5,..., 0 N e e/η e e/η e e/η e e/η e- 8.0e- 9.90e- 9.99e- 9.99e-.0e+0.00e+0.0e e- 5.3e- 9.80e- 9.87e- 9.99e-.00e+0.00e+0.00e e-.86e- 9.43e- 9.53e- 9.99e- 9.99e-.00e+0.00e e-.00e- 8.4e- 8.5e- 9.97e- 9.98e-.00e+0.00e+0 Table 6.4. Uniform mesh: the components η and η 3 of the upper maximum norm error estimator η and the efficiency constant e/η for η. ε = 0 4 ε = 0 7 ε = 0 0 N η η 3 = η e/η η η 3 = η e/η η η 3 = η e/η 6 9.9e- 6.0e+.0e+0 9.9e- 6.0e+5.0e+0 9.9e- 6.0e+8.0e e- 3.e+.00e e- 3.e+5.00e e- 3.e+8.00e e-.56e+ 9.99e- 9.99e-.56e+5.00e e-.56e+8.00e e- 7.8e+ 9.98e-.00e+0 7.8e+4.00e+0.00e+0 7.8e+7.00e+0 We let ε = 0 k, k =,..., 0 and N = k } 9 k=5, with N = N = N 3 = N. Two tensor-product meshes are considered: a variant of the layer-adapted mesh by Bakhvalov [4] and a simple uniform mesh; see Tables For ε ε, our Bakhvalov-type mesh is defined by x [i] = x[i] = x[i] 3 := ϕ(i/n), i = 0,,..., N, where ϕ(t) := ελ ln [b/(b t)] for t [0, θ], ϕ() :=, and ϕ(t) is continuous on [0, ] and linear on [θ, ]. We use the constants b = /, ε = b/λ, and θ = b ελ. The constant λ will be specified later. For ε > ε, the Bakhvalov mesh is defined to be a simple uniform mesh. Note that a suitable Bakhvalov-type layeradapted mesh yields ε-uniform second-order accuracy [4, 8]. Besides, we expect efficient adaptive algorithms to generate meshes that are similar, in some sense, to a Bakhvalov mesh, as in [0, 6 and Figure ]. As a test problem, we use linear problem (3.) with p(x) := and f(x) such that the exact solution is given by u(x) = ( cos( πx ) µ(x ) )( x µ(x ) )( x 3 µ(x 3 ) ), µ(t) = e t/ε e /ε e /ε. Note that this u(x) exhibits boundary and corner layers. In Tables 6. and 6., we give numerical results for the Bakhvalov mesh with λ = 3. Under this choice of λ, the mesh yields ε-uniform second-order accuracy in the maximum norm, so, roughly speaking, one would like to be able to construct similar adaptive meshes. Examining Table 6., we observe agreement with our theoretical estimate (6.). Not only does e/η stabilize, but it becomes close to the linear interpolation error constant /8 =.5e. The components η n of η can be compared when examining Table 6.. For ε = 0 k, k =,..., 0, we observe that η η 3 = η. Furthermore, for ε 0 we have η η η 3, while for ε = 0 the quantity η is dominated by η and η 3. The quantity η 0 is not given, as it is negligible (and known a priori).
17 7 Table 6.5. Bakhvalov mesh, λ = : maximum norm error e, upper maximum norm error estimator η, its components η, η, η 3, and its efficiency constant e/η. ε = 0 5 ε = 0 0 N e η η η 3 = η e/η e η η η 3 = η e/η 6.75e- 5.49e- 3.77e- 4.36e- 4.0e-.4e- 5.49e- 3.77e- 4.83e- 4.64e- 3.0e-.47e-.67e-.3e- 4.8e-.3e-.47e-.58e-.43e- 5.40e e- 3.78e e-3.05e- 5.9e- 7.06e- 3.79e e-3.e- 5.8e- 8.75e- 9.6e-4 4.e-3 5.6e- 5.34e- 3.65e- 9.6e e e- 6.08e- Numerical results for uniform meshes are given in Tables 6.3 and 6.4. On these meshes, the boundary layers are not resolved and e = O(). This is correctly identified by η = η 3 blowing up even more significantly than e. Note that the component η also correctly indicates that the method is inaccurate, but, unlike η 3, it remains bounded. Furthermore, η better reflects the actual errors since e/η const =.0 in Table 6.4. Table 6.5 gives numerical results for the Bakhvalov mesh, but now with λ =. Thus the condition λ >, which implies ε-uniform second-order accuracy for our test problem [4, 8], is violated. Hence the errors slightly increase as ε 0. In this case, we observe that η is too small compared to η and e. In summary, our numerical results suggest that the error estimator η correctly indicates whether or not the method is ε-uniformly accurate. We also note that the quantity η = η 3 may blow up (see Table 6.4), while the component η is sometimes too optimistic (see Table 6.5). The component η seems the most relevant estimator for the actual error e. In particular, η does not blow up, like η 3, and hence seems a more suitable error indicator in possible adaptive mesh construction. We finally note that our conclusions agree with the numerical results in two dimensions [9]. References [] M. Ainsworth and I. Babuška, Reliable and robust a posteriori error estimating for singularly perturbed reaction-diffusion problems, SIAM J. Numer. Anal., 36 (999), [] T. Apel, Anisotropic finite elements: Local estimates and applications, Teubner, Stuttgart, 999. [3] I. Babuška and T. Strouboulis, The finite element method and its reliability, Clarendon Press, Oxford University Press, New York, 00. [4] N.S. Bakhvalov, Towards optimization of methods for solving boundary value problems in the presence of a boundary layer, Zh. Vychisl. Mat. i Mat. Fiz., 9 (969), (in Russian). [5] I. Boglaev, On monotone iterative methods for a nonlinear singularly perturbed reactiondiffusion problem, J. Comput. Appl. Math., 6 (004), [6] M.J. Castro-Diaz, F.F. Hetch, B. Mohammadi and O. Pironneau, Anisotropic unstructed mesh adaptation for flow simulations, Internat. J. Numer. Methods Fluids, 5 (997), [7] L. Chen, P. Sun, and J. Xu, Optimal anisotropic simplicial meshes for minimizing interpolation errors in L p-norm, Math. Comp., 76 (007), [8] C. Clavero, J.L. Gracia and E. Riordan, A parameter robust numerical method for a two dimensional reaction-diffusion problem, Math. Comp., 74 (005), [9] E.F. D Azevedo, Optimal triangular mesh generation by coordinate transformation, SIAM J. Sci. Statist. Comput., (99), [0] E.F. D Azevedo and R.B. Simpson, On optimal interpolation triangle incidences, SIAM J. Sci. Statist. Comput., 0 (989),
18 8 [] V. Dolejší and J. Felcman, Anisotropic mesh adaptation for numerical solution of boundary value problems, Numer. Methods Partial Differential Equations, 0 (004), [] K. Eriksson, An adaptive finite element method with efficient maximum norm error control for elliptic problems, Math. Models Methods Appl. Sci., 4 (994), [3] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 998. [4] J. Guzmán, D. Leykekhman, J. Rossmann and A. H. Schatz, Hölder estimates for Green s functions on convex polyhedral domains and their applications to finite element methods, Numer. Math., (009), 43. [5] W.G. Habashi, J. Dompierre, Y. Bourgault, D. Ait-Ali-Yahia, M. Fortin and M.-G. Vallet, Anistopic mesh adaptation: towards user independent, mesh independent and solver independent CFD. Part I: General principles, Internat. J. Numer. Methods Fluids, 3 (000), [6] N. Kopteva, Maximum norm a posteriori error estimates for a one-dimensional convectiondiffusion problem, SIAM J. Numer. Anal., 39 (00), [7] N. Kopteva, Maximum norm a posteriori error estimates for a one-dimensional singularly perturbed semilinear reaction-diffusion problem, IMA J. Numer. Anal., 7 (007), [8] N. Kopteva, Maximum norm error analysis of a d singularly perturbed semilinear reactiondiffusion problem, Math. Comp., 76 (007), [9] N. Kopteva, Maximum norm a posteriori error estimate for a d singularly perturbed reaction-diffusion problem, SIAM J. Numer. Anal., 46 (008), [0] N. Kopteva and M. Stynes, A robust adaptive method for a quasilinear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (00), [] G. Kunert and R. Verfürth, Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes, Numer. Math., 86 (000), [] G. Kunert, Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes. A posteriori error estimation and adaptive computational methods, Adv. Comput. Math., 5 (00), [3] O.A. Ladyzhenskaya and N.N. Ural tseva, Linear and quasilinear elliptic equations, Academic Press, New York, 968. [4] T. Linß, Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem, BIT, 47 (007), [5] T. Linß and M. Stynes, The SDFEM on Shishkin meshes for linear convection-diffusion problems, Numer. Math., 87 (00), [6] J.J. H. Miller, E. O Riordan and G.I. Shishkin, Solution of Singularly Perturbed Problems with ε-uniform Numerical Methods Introduction to the Theory of Linear Problems in One and Two Dimensions, World Scientific, Singapore, 996. [7] R.H. Nochetto, Pointwise a posteriori error estimates for elliptic problems on highly graded meshes, Math. Comp., 64 (995),. [8] A.D. Polyanin, Handbook of linear partial differential equations for engineers and scientists, CRC Press, London, 00. [9] J. Remacle, X. Li, M.S. Shephard and J.E. Flaherty, Anisotropic adaptive simulation of transient flows using discountinuous Galerkin methods, Internat. J. Numer. Methods Engrg., 6 (005), [30] A.H. Schatz and L.B. Wahlbin, On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Math. Comp., 40 (983),
A PARAMETER ROBUST NUMERICAL METHOD FOR A TWO DIMENSIONAL REACTION-DIFFUSION PROBLEM
A PARAMETER ROBUST NUMERICAL METHOD FOR A TWO DIMENSIONAL REACTION-DIFFUSION PROBLEM C. CLAVERO, J.L. GRACIA, AND E. O RIORDAN Abstract. In this paper a singularly perturbed reaction diffusion partial
More informationRemarks on the analysis of finite element methods on a Shishkin mesh: are Scott-Zhang interpolants applicable?
Remarks on the analysis of finite element methods on a Shishkin mesh: are Scott-Zhang interpolants applicable? Thomas Apel, Hans-G. Roos 22.7.2008 Abstract In the first part of the paper we discuss minimal
More informationSuperconvergence analysis of the SDFEM for elliptic problems with characteristic layers
Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers S. Franz, T. Linß, H.-G. Roos Institut für Numerische Mathematik, Technische Universität Dresden, D-01062, Germany
More informationGreen s function estimates for a singularly perturbed convection-diffusion problem
Green s function estimates for a singularly perturbed convection-diffusion problem Sebastian Franz Natalia Kopteva Abstract We consider a singularly perturbed convection-diffusion problem posed in the
More informationParameter robust methods for second-order complex-valued reaction-diffusion equations
Postgraduate Modelling Research Group, 10 November 2017 Parameter robust methods for second-order complex-valued reaction-diffusion equations Faiza Alssaedi Supervisor: Niall Madden School of Mathematics,
More informationLocal pointwise a posteriori gradient error bounds for the Stokes equations. Stig Larsson. Heraklion, September 19, 2011 Joint work with A.
Local pointwise a posteriori gradient error bounds for the Stokes equations Stig Larsson Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg Heraklion, September
More informationAbstract. 1. Introduction
Journal of Computational Mathematics Vol.28, No.2, 2010, 273 288. http://www.global-sci.org/jcm doi:10.4208/jcm.2009.10-m2870 UNIFORM SUPERCONVERGENCE OF GALERKIN METHODS FOR SINGULARLY PERTURBED PROBLEMS
More informationInterior Layers in Singularly Perturbed Problems
Interior Layers in Singularly Perturbed Problems Eugene O Riordan Abstract To construct layer adapted meshes for a class of singularly perturbed problems, whose solutions contain boundary layers, it is
More informationIterative Domain Decomposition Methods for Singularly Perturbed Nonlinear Convection-Diffusion Equations
Iterative Domain Decomposition Methods for Singularly Perturbed Nonlinear Convection-Diffusion Equations P.A. Farrell 1, P.W. Hemker 2, G.I. Shishkin 3 and L.P. Shishkina 3 1 Department of Computer Science,
More informationASYMPTOTICALLY EXACT A POSTERIORI ESTIMATORS FOR THE POINTWISE GRADIENT ERROR ON EACH ELEMENT IN IRREGULAR MESHES. PART II: THE PIECEWISE LINEAR CASE
MATEMATICS OF COMPUTATION Volume 73, Number 246, Pages 517 523 S 0025-5718(0301570-9 Article electronically published on June 17, 2003 ASYMPTOTICALLY EXACT A POSTERIORI ESTIMATORS FOR TE POINTWISE GRADIENT
More informationA first-order system Petrov-Galerkin discretisation for a reaction-diffusion problem on a fitted mesh
A first-order system Petrov-Galerkin discretisation for a reaction-diffusion problem on a fitted mesh James Adler, Department of Mathematics Tufts University Medford, MA 02155 Scott MacLachlan Department
More informationHamburger Beiträge zur Angewandten Mathematik
Hamburger Beiträge zur Angewandten Mathematik Numerical analysis of a control and state constrained elliptic control problem with piecewise constant control approximations Klaus Deckelnick and Michael
More informationELLIPTIC RECONSTRUCTION AND A POSTERIORI ERROR ESTIMATES FOR PARABOLIC PROBLEMS
ELLIPTIC RECONSTRUCTION AND A POSTERIORI ERROR ESTIMATES FOR PARABOLIC PROBLEMS CHARALAMBOS MAKRIDAKIS AND RICARDO H. NOCHETTO Abstract. It is known that the energy technique for a posteriori error analysis
More informationSPLINE COLLOCATION METHOD FOR SINGULAR PERTURBATION PROBLEM. Mirjana Stojanović University of Novi Sad, Yugoslavia
GLASNIK MATEMATIČKI Vol. 37(57(2002, 393 403 SPLINE COLLOCATION METHOD FOR SINGULAR PERTURBATION PROBLEM Mirjana Stojanović University of Novi Sad, Yugoslavia Abstract. We introduce piecewise interpolating
More informationb i (x) u + c(x)u = f in Ω,
SIAM J. NUMER. ANAL. Vol. 39, No. 6, pp. 1938 1953 c 2002 Society for Industrial and Applied Mathematics SUBOPTIMAL AND OPTIMAL CONVERGENCE IN MIXED FINITE ELEMENT METHODS ALAN DEMLOW Abstract. An elliptic
More informationMaximum-norm stability of the finite element Ritz projection with mixed boundary conditions
Noname manuscript No. (will be inserted by the editor) Maximum-norm stability of the finite element Ritz projection with mixed boundary conditions Dmitriy Leykekhman Buyang Li Received: date / Accepted:
More informationConvergence on Layer-adapted Meshes and Anisotropic Interpolation Error Estimates of Non-Standard Higher Order Finite Elements
Convergence on Layer-adapted Meshes and Anisotropic Interpolation Error Estimates of Non-Standard Higher Order Finite Elements Sebastian Franz a,1, Gunar Matthies b, a Department of Mathematics and Statistics,
More informationFinite Element Superconvergence Approximation for One-Dimensional Singularly Perturbed Problems
Finite Element Superconvergence Approximation for One-Dimensional Singularly Perturbed Problems Zhimin Zhang Department of Mathematics Wayne State University Detroit, MI 48202 Received 19 July 2000; accepted
More informationSuperconvergence Using Pointwise Interpolation in Convection-Diffusion Problems
Als Manuskript gedruckt Technische Universität Dresden Herausgeber: Der Rektor Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems Sebastian Franz MATH-M-04-2012 July 9, 2012
More informationA LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE. 1.
A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE THOMAS CHEN AND NATAŠA PAVLOVIĆ Abstract. We prove a Beale-Kato-Majda criterion
More informationON QUALITATIVE PROPERTIES OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH STRONG DEPENDENCE ON THE GRADIENT
GLASNIK MATEMATIČKI Vol. 49(69)(2014), 369 375 ON QUALITATIVE PROPERTIES OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH STRONG DEPENDENCE ON THE GRADIENT Jadranka Kraljević University of Zagreb, Croatia
More informationCholesky factorisations of linear systems coming from a finite difference method applied to singularly perturbed problems
Cholesky factorisations of linear systems coming from a finite difference method applied to singularly perturbed problems Thái Anh Nhan and Niall Madden The Boundary and Interior Layers - Computational
More informationA Singularly Perturbed Convection Diffusion Turning Point Problem with an Interior Layer
A Singularly Perturbed Convection Diffusion Turning Point Problem with an Interior Layer E. O Riordan, J. Quinn School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland. Abstract
More informationA uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction diffusion problems
IMA Journal of Numerical Analysis (2003 23, 627 644 A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction diffusion problems NIALL MADDEN Department of
More informationBoundary layers in a two-point boundary value problem with fractional derivatives
Boundary layers in a two-point boundary value problem with fractional derivatives J.L. Gracia and M. Stynes Institute of Mathematics and Applications (IUMA) and Department of Applied Mathematics, University
More informationA coupled system of singularly perturbed semilinear reaction-diffusion equations
A coupled system of singularly perturbed semilinear reaction-diffusion equations J.L. Gracia, F.J. Lisbona, M. Madaune-Tort E. O Riordan September 9, 008 Abstract In this paper singularly perturbed semilinear
More informationA uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem
Journal of Modern Methods in Numerical Mathematics 6:1 (2015, 28 43 A uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem
More informationCholesky factorisation of linear systems coming from finite difference approximations of singularly perturbed problems
Cholesky factorisation of linear systems coming from finite difference approximations of singularly perturbed problems Thái Anh Nhan and Niall Madden Abstract We consider the solution of large linear systems
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationRegularity for Poisson Equation
Regularity for Poisson Equation OcMountain Daylight Time. 4, 20 Intuitively, the solution u to the Poisson equation u= f () should have better regularity than the right hand side f. In particular one expects
More informationSUPERCONVERGENCE OF DG METHOD FOR ONE-DIMENSIONAL SINGULARLY PERTURBED PROBLEMS *1)
Journal of Computational Mathematics, Vol.5, No., 007, 185 00. SUPERCONVERGENCE OF DG METHOD FOR ONE-DIMENSIONAL SINGULARLY PERTURBED PROBLEMS *1) Ziqing Xie (College of Mathematics and Computer Science,
More informationACCURATE NUMERICAL METHOD FOR BLASIUS PROBLEM FOR FLOW PAST A FLAT PLATE WITH MASS TRANSFER
FDS-2000 Conference, pp. 1 10 M. Sapagovas et al. (Eds) c 2000 MII ACCURATE UMERICAL METHOD FOR BLASIUS PROBLEM FOR FLOW PAST A FLAT PLATE WITH MASS TRASFER B. GAHA 1, J. J. H. MILLER 2, and G. I. SHISHKI
More informationAn exponentially graded mesh for singularly perturbed problems
An exponentially graded mesh for singularly perturbed problems Christos Xenophontos Department of Mathematics and Statistics University of Cyprus joint work with P. Constantinou (UCY), S. Franz (TU Dresden)
More informationA FAST SOLVER FOR ELLIPTIC EQUATIONS WITH HARMONIC COEFFICIENT APPROXIMATIONS
Proceedings of ALGORITMY 2005 pp. 222 229 A FAST SOLVER FOR ELLIPTIC EQUATIONS WITH HARMONIC COEFFICIENT APPROXIMATIONS ELENA BRAVERMAN, MOSHE ISRAELI, AND ALEXANDER SHERMAN Abstract. Based on a fast subtractional
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationPIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED
PIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED ALAN DEMLOW Abstract. Recent results of Schatz show that standard Galerkin finite element methods employing piecewise polynomial elements of degree
More informationLocal Mesh Refinement with the PCD Method
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 1, pp. 125 136 (2013) http://campus.mst.edu/adsa Local Mesh Refinement with the PCD Method Ahmed Tahiri Université Med Premier
More informationRobust exponential convergence of hp-fem for singularly perturbed systems of reaction-diffusion equations
Robust exponential convergence of hp-fem for singularly perturbed systems of reaction-diffusion equations Christos Xenophontos Department of Mathematics and Statistics University of Cyprus joint work with
More informationarxiv: v2 [math.na] 23 Apr 2016
Improved ZZ A Posteriori Error Estimators for Diffusion Problems: Conforming Linear Elements arxiv:508.009v2 [math.na] 23 Apr 206 Zhiqiang Cai Cuiyu He Shun Zhang May 2, 208 Abstract. In [8], we introduced
More informationTong Sun Department of Mathematics and Statistics Bowling Green State University, Bowling Green, OH
Consistency & Numerical Smoothing Error Estimation An Alternative of the Lax-Richtmyer Theorem Tong Sun Department of Mathematics and Statistics Bowling Green State University, Bowling Green, OH 43403
More informationParameter-Uniform Numerical Methods for a Class of Singularly Perturbed Problems with a Neumann Boundary Condition
Parameter-Uniform Numerical Methods for a Class of Singularly Perturbed Problems with a Neumann Boundary Condition P. A. Farrell 1,A.F.Hegarty 2, J. J. H. Miller 3, E. O Riordan 4,andG.I. Shishkin 5 1
More informationSECOND ORDER TIME DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS
Proceedings of ALGORITMY 2009 pp. 1 10 SECOND ORDER TIME DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS MILOSLAV VLASÁK Abstract. We deal with a numerical solution of a scalar
More informationfor all subintervals I J. If the same is true for the dyadic subintervals I D J only, we will write ϕ BMO d (J). In fact, the following is true
3 ohn Nirenberg inequality, Part I A function ϕ L () belongs to the space BMO() if sup ϕ(s) ϕ I I I < for all subintervals I If the same is true for the dyadic subintervals I D only, we will write ϕ BMO
More informationS chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1.
Sep. 1 9 Intuitively, the solution u to the Poisson equation S chauder Theory u = f 1 should have better regularity than the right hand side f. In particular one expects u to be twice more differentiable
More informationAdaptive methods for control problems with finite-dimensional control space
Adaptive methods for control problems with finite-dimensional control space Saheed Akindeinde and Daniel Wachsmuth Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy
More informationA Proximal Method for Identifying Active Manifolds
A Proximal Method for Identifying Active Manifolds W.L. Hare April 18, 2006 Abstract The minimization of an objective function over a constraint set can often be simplified if the active manifold of the
More informationA Linearised Singularly Perturbed Convection- Diffusion Problem with an Interior Layer
Dublin Institute of Technology ARROW@DIT Articles School of Mathematics 201 A Linearised Singularly Perturbed Convection- Diffusion Problem with an Interior Layer Eugene O'Riordan Dublin City University,
More informationThe continuity method
The continuity method The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to elliptic partial differential equations. One crucial
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 37, pp. 166-172, 2010. Copyright 2010,. ISSN 1068-9613. ETNA A GRADIENT RECOVERY OPERATOR BASED ON AN OBLIQUE PROJECTION BISHNU P. LAMICHHANE Abstract.
More informationA NOTE ON THE LADYŽENSKAJA-BABUŠKA-BREZZI CONDITION
A NOTE ON THE LADYŽENSKAJA-BABUŠKA-BREZZI CONDITION JOHNNY GUZMÁN, ABNER J. SALGADO, AND FRANCISCO-JAVIER SAYAS Abstract. The analysis of finite-element-like Galerkin discretization techniques for the
More informationThree remarks on anisotropic finite elements
Three remarks on anisotropic finite elements Thomas Apel Universität der Bundeswehr München Workshop Numerical Analysis for Singularly Perturbed Problems dedicated to the 60th birthday of Martin Stynes
More information[2] (a) Develop and describe the piecewise linear Galerkin finite element approximation of,
269 C, Vese Practice problems [1] Write the differential equation u + u = f(x, y), (x, y) Ω u = 1 (x, y) Ω 1 n + u = x (x, y) Ω 2, Ω = {(x, y) x 2 + y 2 < 1}, Ω 1 = {(x, y) x 2 + y 2 = 1, x 0}, Ω 2 = {(x,
More informationA RIEMANN PROBLEM FOR THE ISENTROPIC GAS DYNAMICS EQUATIONS
A RIEMANN PROBLEM FOR THE ISENTROPIC GAS DYNAMICS EQUATIONS KATARINA JEGDIĆ, BARBARA LEE KEYFITZ, AND SUN CICA ČANIĆ We study a Riemann problem for the two-dimensional isentropic gas dynamics equations
More informationOn Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations
On Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations G. Seregin, V. Šverák Dedicated to Vsevolod Alexeevich Solonnikov Abstract We prove two sufficient conditions for local regularity
More informationAuthor(s) Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 41(1)
Title On the stability of contact Navier-Stokes equations with discont free b Authors Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 4 Issue 4-3 Date Text Version publisher URL
More informationLECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES. Sergey Korotov,
LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES Sergey Korotov, Institute of Mathematics Helsinki University of Technology, Finland Academy of Finland 1 Main Problem in Mathematical
More informationON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS
Chin. Ann. Math.??B(?), 200?, 1 20 DOI: 10.1007/s11401-007-0001-x ON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS Michel CHIPOT Abstract We present a method allowing to obtain existence of a solution for
More informationA Multigrid Method for Two Dimensional Maxwell Interface Problems
A Multigrid Method for Two Dimensional Maxwell Interface Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University USA JSA 2013 Outline A
More informationError estimates for the Raviart-Thomas interpolation under the maximum angle condition
Error estimates for the Raviart-Thomas interpolation under the maximum angle condition Ricardo G. Durán and Ariel L. Lombardi Abstract. The classical error analysis for the Raviart-Thomas interpolation
More informationSYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy
More informationA Posteriori Existence in Adaptive Computations
Report no. 06/11 A Posteriori Existence in Adaptive Computations Christoph Ortner This short note demonstrates that it is not necessary to assume the existence of exact solutions in an a posteriori error
More informationOn angle conditions in the finite element method. Institute of Mathematics, Academy of Sciences Prague, Czech Republic
On angle conditions in the finite element method Michal Křížek Institute of Mathematics, Academy of Sciences Prague, Czech Republic Joint work with Jan Brandts (University of Amsterdam), Antti Hannukainen
More informationDissipative quasi-geostrophic equations with L p data
Electronic Journal of Differential Equations, Vol. (), No. 56, pp. 3. ISSN: 7-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Dissipative quasi-geostrophic
More informationDeforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary
Deforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary Weimin Sheng (Joint with Li-Xia Yuan) Zhejiang University IMS, NUS, 8-12 Dec 2014 1 / 50 Outline 1 Prescribing
More informationABOUT SOME TYPES OF BOUNDARY VALUE PROBLEMS WITH INTERFACES
13 Kragujevac J. Math. 3 27) 13 26. ABOUT SOME TYPES OF BOUNDARY VALUE PROBLEMS WITH INTERFACES Boško S. Jovanović University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11 Belgrade, Serbia
More informationYongdeok Kim and Seki Kim
J. Korean Math. Soc. 39 (00), No. 3, pp. 363 376 STABLE LOW ORDER NONCONFORMING QUADRILATERAL FINITE ELEMENTS FOR THE STOKES PROBLEM Yongdeok Kim and Seki Kim Abstract. Stability result is obtained for
More information2 A Model, Harmonic Map, Problem
ELLIPTIC SYSTEMS JOHN E. HUTCHINSON Department of Mathematics School of Mathematical Sciences, A.N.U. 1 Introduction Elliptic equations model the behaviour of scalar quantities u, such as temperature or
More informationMaximum-norm a posteriori estimates for discontinuous Galerkin methods
Maximum-norm a posteriori estimates for discontinuous Galerkin methods Emmanuil Georgoulis Department of Mathematics, University of Leicester, UK Based on joint work with Alan Demlow (Kentucky, USA) DG
More informationUltraconvergence of ZZ Patch Recovery at Mesh Symmetry Points
Ultraconvergence of ZZ Patch Recovery at Mesh Symmetry Points Zhimin Zhang and Runchang Lin Department of Mathematics, Wayne State University Abstract. The ultraconvergence property of the Zienkiewicz-Zhu
More informationA collocation method for solving some integral equations in distributions
A collocation method for solving some integral equations in distributions Sapto W. Indratno Department of Mathematics Kansas State University, Manhattan, KS 66506-2602, USA sapto@math.ksu.edu A G Ramm
More informationAn Accurate Fourier-Spectral Solver for Variable Coefficient Elliptic Equations
An Accurate Fourier-Spectral Solver for Variable Coefficient Elliptic Equations Moshe Israeli Computer Science Department, Technion-Israel Institute of Technology, Technion city, Haifa 32000, ISRAEL Alexander
More informationGoal. Robust A Posteriori Error Estimates for Stabilized Finite Element Discretizations of Non-Stationary Convection-Diffusion Problems.
Robust A Posteriori Error Estimates for Stabilized Finite Element s of Non-Stationary Convection-Diffusion Problems L. Tobiska and R. Verfürth Universität Magdeburg Ruhr-Universität Bochum www.ruhr-uni-bochum.de/num
More informationNodal O(h 4 )-superconvergence of piecewise trilinear FE approximations
Preprint, Institute of Mathematics, AS CR, Prague. 2007-12-12 INSTITTE of MATHEMATICS Academy of Sciences Czech Republic Nodal O(h 4 )-superconvergence of piecewise trilinear FE approximations Antti Hannukainen
More informationComm. Nonlin. Sci. and Numer. Simul., 12, (2007),
Comm. Nonlin. Sci. and Numer. Simul., 12, (2007), 1390-1394. 1 A Schrödinger singular perturbation problem A.G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA ramm@math.ksu.edu
More informationBEST APPROXIMATION PROPERTY IN THE W FINITE ELEMENT METHODS ON GRADED MESHES.
BEST APPROXIMATION PROPERTY IN THE W 1 NORM FOR FINITE ELEMENT METHODS ON GRADED MESHES. A. DEMLOW, D. LEYKEKHMAN, A.H. SCHATZ, AND L.B. WAHLBIN Abstract. We consider finite element methods for a model
More informationTraces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis
More informationMaximum norm estimates for energy-corrected finite element method
Maximum norm estimates for energy-corrected finite element method Piotr Swierczynski 1 and Barbara Wohlmuth 1 Technical University of Munich, Institute for Numerical Mathematics, piotr.swierczynski@ma.tum.de,
More informationOn solving linear systems arising from Shishkin mesh discretizations
On solving linear systems arising from Shishkin mesh discretizations Petr Tichý Faculty of Mathematics and Physics, Charles University joint work with Carlos Echeverría, Jörg Liesen, and Daniel Szyld October
More informationA CHARACTERIZATION OF STRICT LOCAL MINIMIZERS OF ORDER ONE FOR STATIC MINMAX PROBLEMS IN THE PARAMETRIC CONSTRAINT CASE
Journal of Applied Analysis Vol. 6, No. 1 (2000), pp. 139 148 A CHARACTERIZATION OF STRICT LOCAL MINIMIZERS OF ORDER ONE FOR STATIC MINMAX PROBLEMS IN THE PARAMETRIC CONSTRAINT CASE A. W. A. TAHA Received
More informationENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS
ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS CARLO LOVADINA AND ROLF STENBERG Abstract The paper deals with the a-posteriori error analysis of mixed finite element methods
More informationSimple Examples on Rectangular Domains
84 Chapter 5 Simple Examples on Rectangular Domains In this chapter we consider simple elliptic boundary value problems in rectangular domains in R 2 or R 3 ; our prototype example is the Poisson equation
More informationThe De Giorgi-Nash-Moser Estimates
The De Giorgi-Nash-Moser Estimates We are going to discuss the the equation Lu D i (a ij (x)d j u) = 0 in B 4 R n. (1) The a ij, with i, j {1,..., n}, are functions on the ball B 4. Here and in the following
More informationIntroduction. J.M. Burgers Center Graduate Course CFD I January Least-Squares Spectral Element Methods
Introduction In this workshop we will introduce you to the least-squares spectral element method. As you can see from the lecture notes, this method is a combination of the weak formulation derived from
More informationERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX
ERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX JOHN LOFTIN, SHING-TUNG YAU, AND ERIC ZASLOW 1. Main result The purpose of this erratum is to correct an error in the proof of the main result
More informationPolynomial Preserving Recovery for Quadratic Elements on Anisotropic Meshes
Polynomial Preserving Recovery for Quadratic Elements on Anisotropic Meshes Can Huang, 1 Zhimin Zhang 1, 1 Department of Mathematics, Wayne State University, Detroit, Michigan 480 College of Mathematics
More informationA posteriori error estimates for non conforming approximation of eigenvalue problems
A posteriori error estimates for non conforming approximation of eigenvalue problems E. Dari a, R. G. Durán b and C. Padra c, a Centro Atómico Bariloche, Comisión Nacional de Energía Atómica and CONICE,
More informationand finally, any second order divergence form elliptic operator
Supporting Information: Mathematical proofs Preliminaries Let be an arbitrary bounded open set in R n and let L be any elliptic differential operator associated to a symmetric positive bilinear form B
More informationξ,i = x nx i x 3 + δ ni + x n x = 0. x Dξ = x i ξ,i = x nx i x i x 3 Du = λ x λ 2 xh + x λ h Dξ,
1 PDE, HW 3 solutions Problem 1. No. If a sequence of harmonic polynomials on [ 1,1] n converges uniformly to a limit f then f is harmonic. Problem 2. By definition U r U for every r >. Suppose w is a
More informationApproximation by Conditionally Positive Definite Functions with Finitely Many Centers
Approximation by Conditionally Positive Definite Functions with Finitely Many Centers Jungho Yoon Abstract. The theory of interpolation by using conditionally positive definite function provides optimal
More informationAn a posteriori error estimate and a Comparison Theorem for the nonconforming P 1 element
Calcolo manuscript No. (will be inserted by the editor) An a posteriori error estimate and a Comparison Theorem for the nonconforming P 1 element Dietrich Braess Faculty of Mathematics, Ruhr-University
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationPREPRINT 2010:23. A nonconforming rotated Q 1 approximation on tetrahedra PETER HANSBO
PREPRINT 2010:23 A nonconforming rotated Q 1 approximation on tetrahedra PETER HANSBO Department of Mathematical Sciences Division of Mathematics CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG
More informationOn an Approximation Result for Piecewise Polynomial Functions. O. Karakashian
BULLETIN OF THE GREE MATHEMATICAL SOCIETY Volume 57, 010 (1 7) On an Approximation Result for Piecewise Polynomial Functions O. arakashian Abstract We provide a new approach for proving approximation results
More informationOblique derivative problems for elliptic and parabolic equations, Lecture II
of the for elliptic and parabolic equations, Lecture II Iowa State University July 22, 2011 of the 1 2 of the of the As a preliminary step in our further we now look at a special situation for elliptic.
More informationRobust error estimates for regularization and discretization of bang-bang control problems
Robust error estimates for regularization and discretization of bang-bang control problems Daniel Wachsmuth September 2, 205 Abstract We investigate the simultaneous regularization and discretization of
More informationMultigrid finite element methods on semi-structured triangular grids
XXI Congreso de Ecuaciones Diferenciales y Aplicaciones XI Congreso de Matemática Aplicada Ciudad Real, -5 septiembre 009 (pp. 8) Multigrid finite element methods on semi-structured triangular grids F.J.
More informationAnalysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods
Advances in Applied athematics and echanics Adv. Appl. ath. ech., Vol. 1, No. 6, pp. 830-844 DOI: 10.408/aamm.09-m09S09 December 009 Analysis of Two-Grid ethods for Nonlinear Parabolic Equations by Expanded
More informationExplosive Solution of the Nonlinear Equation of a Parabolic Type
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 5, 233-239 Explosive Solution of the Nonlinear Equation of a Parabolic Type T. S. Hajiev Institute of Mathematics and Mechanics, Acad. of Sciences Baku,
More informationarxiv: v1 [math.ap] 18 Jan 2019
manuscripta mathematica manuscript No. (will be inserted by the editor) Yongpan Huang Dongsheng Li Kai Zhang Pointwise Boundary Differentiability of Solutions of Elliptic Equations Received: date / Revised
More informationDISCRETE MAXIMUM PRINCIPLES in THE FINITE ELEMENT SIMULATIONS
DISCRETE MAXIMUM PRINCIPLES in THE FINITE ELEMENT SIMULATIONS Sergey Korotov BCAM Basque Center for Applied Mathematics http://www.bcamath.org 1 The presentation is based on my collaboration with several
More information