SPLINE COLLOCATION METHOD FOR SINGULAR PERTURBATION PROBLEM. Mirjana Stojanović University of Novi Sad, Yugoslavia
|
|
- Adele Malone
- 6 years ago
- Views:
Transcription
1 GLASNIK MATEMATIČKI Vol. 37(57(2002, SPLINE COLLOCATION METHOD FOR SINGULAR PERTURBATION PROBLEM Mirjana Stojanović University of Novi Sad, Yugoslavia Abstract. We introduce piecewise interpolating polynomials as an approximation for the driving terms in the numerical solution of the singularly perturbed differential equation. In this way we obtain the difference scheme which is second order accurate in uniform norm. We verify the convergence rate of presented scheme by numerical experiments. 1. Introduction There are a lot of difference schemes trying to solve thin-layer phenomena in 1D and 2D. The layers are caused by a dominating convection or advection term, or by singularly perturbed behaviour. The aim is to find better numerical techniques for singular perturbation problems, convection-dominated flows, reaction-diffusion problems, etc. A number of new books appear on that subject ([8, 9, 13]. Further development of collocation techniques are towards PDEs singularly perturbed in 2D. The study of numerical behaviour of the solution of a nonlinear reaction diffusion equations with one source term (heat equation arise in plasma physics for the computation. Sufficient conditions of blow up is obtained for the numerical solution as for the exact solution in [14]. The stability and the convergence of the scheme is proved. ɛ-uniform numerical methods are constructed for a class of semilinear problems by classical finite difference operator on special piecewise-uniform meshes (cf. [6]. New direction in application of splines in solving singularly perturbed problems is spline-wavelet decomposition of corresponding Sobolev spaces 2000 Mathematics Subject Classification. 65L10. Key words and phrases. Singularly perturbed self-adjoint problem, interpolating splines, collocation method, uniform convergence in a small parameter. 393
2 394 M. STOJANOVIĆ based on a special point value vanishing property of basis functions, a construction of fast discrete wavelet transform, using collocation method (cf. [3]. The former approaches include fitted finite difference methods, finite element methods using exponential elements and method which uses a priori refined or special meshes. Development of these fields goes to different schemes for boundary layer away from boundary layer with rectangular meshes and their combinations in 2D. Also, the fitted finite difference operators on piecewise-uniform, say, Shishkin meshes are used in [7]. Then, the solutions with two sharp layers are considered: boundary layer and spike layer solution (cf. [4]. Singularly perturbed PDEs are all pervasive in applications of mathematics to problems in the sciences and engineering, say Navier-Stokes equations of fluid flow at high Reynolds number, the drift-diffusion equations of semiconductor device physics, etc. Classical methods are inadequate for solving those problems when ɛ is very small. The main point in the numerical computations in 2D appear to exibit blow up solutions or what is the most important global solutions of the Euler and Navier-Stokes equations, but extreme numerical instability appear near the blow up time and it is difficult to find reliable conclusions. The above results are the subject of [2]. Thus, the numerical solutions of Navier-Stokes and Euler equations are the challenges of new millennium. It is necessary to make substantial progress in that field. Thus, the new directions in solving boundary value problems are PDEs in 2D or more in 3D and new techniques more adequate for very small ɛ. In this paper we revise an old technique spline collocation technique for singular perturbation problem in 1D to obtain optimal difference scheme in uniform norm, with respect to small parameter ɛ. O Riordan and Stynes in [10, 11, 12, 15] are introduced piecewise constants on each subinterval [x i 1, x i ] as an approximation for the functions p(x and f(x in singularly perturbed differential equation in one dimension. In this paper we use quadratic interpolating splines instead of piecewise constants as an approximation for function f(x. 2. Scheme generation Consider the following two-point boundary value problem (1 Lu ɛu + p(xu = f(x, u(0 = α 0, u(1 = α 1 0 < ɛ << 1, α 0, α 1 are given constants, functions p(x and f(x are smooth enough and satisfy p(x β > 0 for x [0, 1]. Then, the solution u(x has a boundary layer at both end points [0, 1].
3 SPLINE COLLOCATION METHOD 395 We seek solution to (1 on each subinterval (x i 1, x i of interval [0, 1] as a solution of differential equation (2 LS (x ɛs (x + p is (x = f i, x I i = (x i 1, x i, S (x i 1 = u i 1, S (x i = u i. The value u i 1 and u i will be determined from the corresponding difference scheme. The corresponding difference scheme is a consequence of the continuity condition of the first derivative of spline at observed points. This method is in principal of collocation type. As an approximation for the function p(x we use piecewise constants p = ( p i 1 + p i /2 where p i = p(x i. For the function f(x we introduce quadratic interpolating spline at points x i 1, x i 1/2 and x i (x i 1/2 is the mid-point of subinterval [x i 1, x i ]: (3 Spline ( f i = f i 1 (2/h 2 (x x i 1/2 (x x i + f i 2/h 2 (x x i 1/2 (x x i 1 ( + f i 1/2 4/h 2 (x x i 1 (x x i. S (x= 1/ sinh ρ i ( sinh ( p i /ɛ(x x i 1 (u i 4f i 1 /(p i ρ 2 i f i(1/p i + 4/(p i ρ 2 i + f i 1/2/p i (8/ρ 2 i (4 sinh ( p i /ɛ(x x i (u i 1 f i 1 (4/(p i ρ 2 i + 1/p i f i (4/(p i ρ 2 i + f i 1/28/(p i ρ 2 i + 2x 2 /(p i h 2 (f i 1 + f i 2f i 1/2 ( + 2x/(p i h 2 f i 1 (x i 1/2 + x i f i (x i 1/2 + x i 1 + 2f i 1/2 (x i 1 + x i + C where ρ i = p i /ɛh, and p i = ( p(x i 1 + p(x i /2 is the solution to (2 with (3. Using continuity condition of (4 at the point x i we obtain spline collocation method which gives the following spline difference scheme where R h v i = Q h f i, i = 1(1n 1, v 0 = α 0, v n = α n, R h v i = r i v i 1 + r c i v i + r + i v i+1, Q h f i = q i f i 1 + q c i f i + q + i f i+1 + q i1/2 f i 1/2 + q + i1/2 f i+1/2.
4 396 M. STOJANOVIĆ We can modify these piecewise constants in order to obtain uniform scheme in the following way: (5 r i = ρ i / sinh ρ i, r + i = ρ i+1 / sinh ρ i+1, r c i = ρ i coth ρ i + ρ i+1 coth ρ i+1, q i = 1/ p i 1 (4/ρ i tanh ρ i /2 ρ i / sinh ρ i 1, q + i = 1/ p i+1 (4/ρ i+1 tanh ρ i+1 /2 ρ i+1 / sinh ρ i+1 1, q c i = 1/ p i (4/ρ i tanh (ρ i /2 + ρ i coth ρ i + 4/ρ i+1 tanh (ρ i+1 /2 + ρ i+1 coth ρ i+1 6, q i1/2 = 1/ p i 1/2( 8/ρ i tanh (ρ i/2 + 4, q + i1/2 = 1/ p i+1/2( 8/ρ i+1 tanh (ρ i+1 / In the sequel we will use the notation O(h i, i = 0(16, to denote a quantity bounded in absolute value by Ch i. 3. Proof of the uniform convergence The proof of the uniform convergence will be given for the case p(x = const, the first. The scheme (5 has the matrix form A V = F where V = [v 1,..., v n 1 ] T and F is also vector. Then, (6 u(x i v i M A 1 max τ i (u, i where τ i (u is the truncated error of the scheme. Lemma 1 ([5]. Let u(x C 4 ([0, 1], and p (0 = p (1 = 0. Then, the solution to (1 has the form (7 u(x = u 0 (x + w 0 (x + g(x where (8 u 0 (x = p 0 exp( x p(0/ɛ (9 w 0 (x = p 1 exp( (1 x p(1/ɛ, p 0, p 1 are bounded functions of ɛ independent of x and (10 g (i (x M(1 + ɛ 1 i/2, i = 0(1n, M is a constant independent of ɛ. Lemma 2. Let p(x = const. Then, the truncated error for the boundary layer functions u o and w 0 equals zero.
5 SPLINE COLLOCATION METHOD 397 Proof. From (2 it is obvious that the approximation for the function f(x affects only the particular solution to (2 and the solution of the homogeneous problem is independent of f(x. The truncated error for the boundary layer function (8 is τ i (u = R h u i Q( h (L(u i. Denote these parts of difference by τ r and τ q. We have τ r = u oi exp(ρ 0 r i + ri c + exp( ρ 0r + i where u 0i = exp( ( p(0/ɛx i, τ q = u 0i (ρ 2 0 ρ 2 i 1 ρ2 i 1 exp(ρ 0qi + (ρ 2 0 ρ2 i ρ2 i qc i + (ρ 2 0 ρ2 i+1 ρ2 i+1 q+ i exp( ρ 0. When p = const part τ q = 0. Using ri, rc i, r+ i from (5 by simple calculation we obtain τ r = 0. It implies τ i (u = 0. Similarly τ i (w 0 = 0. Lemma 3. Estimate of the matrix for the scheme (5 is { (11 A 1 Mɛ/h 2 for h 2 ɛ M ɛ/h for h 2 ɛ, where denotes the usual max norm. Proof. Follows from A 1 = (r i + r c i + r+ i 1 = ( ρ i / sinh ρ i + ρ i coth ρ i + ρ i+1 coth ρ i+1 ρ i+1 / sinh ρ i+1 1. Theorem 4. Let p(x = const and f C 2 ([0, 1]. Let {v i }, i = 1(1n 1 be the approximation of the solution to (1 obtained by (5. Then, there is a constant C independent of ɛ and h such that u(x i v i Ch 3 min(h/ɛ, 1/ ɛ. Proof. From Lemma 2 we have τ i (u = τ i (w for p(x = const. Thus, we have to estimate only function g(x. According to [1] (12 τ i (g = τ (0 i g i + τ (1 i g (1 i + τ (2 i g (2 i + τ (3 i g (3 i + τ (4 i g (4 i R. For the scheme (5 τ (0 i = τ (1 i = τ (2 i = τ (3 i = 0. It remains to estimate τ (4 i. We have τ (4 i = h 4 /24 (r + i + r i p(q i + q + i 1/16(q i1/2 + q+i1/2 + ɛh 2 /p (qi + q i + + 1/4(q +i1/2 + q i1/2. After ordering these expressions we obtain ( τ (4 i = h 4 /24 7/ρ tanh (ρ/2 + 3/2 + ɛh 2( 2/ρ sinh (ρ/2 ρ/ sinh (ρ/2, where ρ = p/ɛh, for p = const. When h 2 ɛ then τ (4 i Mh 4 ρ 2 and for h 2 ɛ we obtain τ (4 i M(h 4 + ɛh 2, i.e. τ (4 i Mh 4. Using (10, (11
6 398 M. STOJANOVIĆ and (6 we obtain u i v i Mh 3 min(h/ɛ, 1/ ɛ. Similarly we can estimate τ (4 i and the remainder terms. They are of the minor order. We have ( ( r i + ri p i 1 qi + p i+1 q i + + 1/32(q +i1/2 q i1/2 ( + ɛh 3 /6 q i + q + i + 1/8( q i1/2 + q+i1/2. τ (5 i = h 5 /5! For p(x = const we have τ (5 i = 0. Then, τ (6 i = h 6 /6! (r i + + ri (p i 1 qi + p i+1 q i + + 1/64(p i 1/2 q i1/2 + p i+1/2q +i1/2 + ɛh 4 /24 (q i + q + i + 1/16(q i1/2 + q+i1/2 and τ (6 i = h 6 /6!( 31/(4ρ tanh (ρ/2 17/8 + ɛh 4 /24(7/ρ tanh (ρ/2 2ρ/ sinh ρ 3/2. When h 2 ɛ we have τ (6 i Mh 6 ρ 2 and for h 2 ɛ we have τ (6 i h 6. Thus, τ (6 i g (6 Mh 6 /ɛ 2 ρ 2 for h 2 ɛ, and τ (6 i g (6 Mh 6 /ɛ 2 for h 2 ɛ. Theorem 5. Let p, f C 2 ([0, 1] and p (0 = p (1 = 0 and {v i }, i = 1(1n 1 be the approximation to (1 obtained by (5. Then, the following estimate holds: u(x i v i Mh min(h, ɛ. Proof. Nodal errors due to function g(x. For the function g(x we have τ (0 i = τ (1 i = 0 and τ (2 i = τ (2 i ( ρ i + (ρ i ρ i + ρ i+1 ρ i ( h 2 a bɛ/p i + C, where ρ = p/ɛh and τ (2 i ( ρ i = 0 (cf. Theorem 4 when p(x = const. We have ρ i ρ + ρ i+1 ρ Mh 3 / ɛ, ρ i / sinh ρ i = ρ/ sinh ρ + (ρ i ρb/ρ i tanh (ρ i /2 = 1/ ρ tanh ( ρ/2 + (ρ i ρa, ρ i coth ρ i = ρ coth ρ + (ρ i ρc where ( b = (sinh ρ ρ cosh ρ/ sinh 2 ρ = ρ/3 + O(ρ 2, a = 1/( ρ coth ( ρ/2 2 coth ( ρ/2 ρ/2(coth 2 ( ρ/2 1 = ρ/12 + O(ρ 2, ( and c = coth ρ ρ(coth 2 ρ 1 = ρ/3+o(ρ 2. So we obtain τ (2 i Mh 4 /ɛ for h 2 ɛ, τ (2 i Mh 2 for h 2 ɛ. Using matrix estimate (11 we obtain the nodal errors due to the function g(x. (13 u(x i v i Mh min(h, ɛ.
7 SPLINE COLLOCATION METHOD 399 [1] Nodal errors due to boundary layer function u 0 (x. According to τ r = u 0i (r i exp(ρ 0 + ri c + r + i exp( ρ 0 and ( τ q = (p 0 /ɛ (p 0 p i 1 q i exp(ρ 0 + (p 0 p i qi c + (p 0 p i+1 q + i exp( ρ 0 + (p 0 p i 1/2 exp(ρ 0 /2q i1/2 + (p 0 p i+1/2 exp( ρ 0 /2q + i1/2. Setting the coefficients of (5 we obtain (14 τ r = 2u 0i ρ i / sinh ρ i (cosh ρ 0 cosh ρ i + O(h 4 /ɛ = v i h 2 /ɛ(p 0 p i + O(h 4 /ɛ and ( ( τ q = (p 0 /ɛ (p 0 p i q i exp(ρ 0 + qi c + q + i exp( ρ 0 + q i1/2 exp(ρ 0/2 + q i1/2 exp( ρ 0 /2 + (p i p i 1 q i exp(ρ 0 + (p i p i+1 q + i exp( ρ 0 + (p i p i 1/2 q i1/2 exp(ρ 0/2 + (p i p i+1/2 q + i1/2 exp( ρ 0/2. Divide τ q into two parts τ q = τ q A + τ q B where (15 τ q A = (p 0 /ɛ(p 0 p i /p i ((4/ρ i tanh (ρ i /2 ρ/ sinh ρ i 12 cosh ρ 0 + (8/ρ i tanh (ρ i /2 + 2ρ i coth ρ i cosh (ρ 0 /2( 8/ρ i tanh (ρ i / N = (p 0 /ɛ(p 0 p i /p i (2A cosh ρ 0 + B + 2C cosh (ρ 0 /2. Here N denotes the part of τ q which is negligible, i.e. it does not influence to the order of uniform convergence which is O(h 2 what is our intention to prove. We denote by A = 4/ρ i tanh (ρ i /2 ρ i / sinh ρ i 1, B = 8/ρ i tanh (ρ i /2 + 2ρ i coth ρ i 6, C = 8/ρ i tanh (ρ i / In a case h 2 ɛ we shall estimate these parts separately. Taylor s expansion gives A = 1/360ρ 4 i + O(ρ5 i, B = ρ2 i /3 ρ4 i /90 + O(ρ6 i, C = ρ2 i /3 ρ4 i /30 + O(ρ 6 i. Putting it in (15 and after ordering the terms we obtain (16 τ q A = (p 0 p i h 2 /ɛ + O(h 4 /ɛ. Subtracting (14 and (15 we obtain (17 τ r τ q A Mh 4 /ɛ for h 2 ɛ.
8 400 M. STOJANOVIĆ Further, we have τ q = u 0i ((p i p i 1 q i exp(ρ 0 + (p i p i+1 q + i exp( ρ 0 + (p i p i 1/2 exp(ρ 0 /2q i1/2 + (p i p i+1/2 q + i1/2 exp( ρ 0/2. Divide this into two parts: D = u 0i ((p i p i 1 qi exp(ρ 0 + (p i p i+1 q i + exp( ρ 0, E = u 0i ((p i p i 1/2 q i1/2 exp(ρ 0/2 + (p 0 p i+1/2 exp( ρ 0 /2. As q i Mρ4 i, q+ i Mρ4 i we obtain D Mh6 /ɛ 2, E = h/2p (ξ 1 exp(ρ 0 /2q i1/2 h/2p (ξ 2 q + i1/2 exp( ρ 0/2 where x i 1 ξ 1 x i, x i ξ 2 x i+1, and E h 2 /2p (ξρ 2 i /3, where ξ 1 ξ ξ 2. It implies E Mh 4 /ɛ. The final estimate is τ q B Mh 4 /ɛ for h 2 ɛ. This estimate, (17 and the matrix estimate give u(x i v i Mh 2 for h 2 ɛ for boundary layer term. In the case h 2 ɛ we have qi M, q+ i M, qc i Mh/ ɛ, q i1/2 M, q + i1/2 M. Thus, τ q (p 0 p i 1 Mu 0i + (p p i h/ ɛmu 0i + (p 0 p i 1/2 Mu 0i + (p 0 p i+1/2 Mu 0i. Using estimate p(0 p(x i Mx 2 i under the condition p (0 = 0 for τ q we obtain (18 τ q Mɛ for h 2 ɛ. In the estimate of τ r we shall find the deviation from τ r ( ρ where ρ = p/ɛh where p = const. Then, τ r = τ r τ r ( ρ = u 0i ((r( ρ r i exp(ρ 0+(ri c ( ρ ri c +(r + i ( ρ r+ i exp( ρ 0. As r + i ( ρ r+ i Mx2 i+1, r i ( ρ r i Mx2 i 1, rc i ( ρ rc i Mx2 i we have (19 τ r Mɛ for h 2 ɛ. We obtain the same estimate for (20 τ i (w Mɛ for h 2 ɛ. Then, (18, (19, (20 and the matrix estimate (11 give (21 u(x i v i Mh ɛ for h 2 ɛ. Thus, (13, (21 and the similar estimate for the function w(x show that the boundary layer term contributes O(h min(h, ɛ to the nodal errors.
9 SPLINE COLLOCATION METHOD Numerical experiments To illustrate computationally the convergence of the presented scheme we consider the following singular two-point boundary value problem (22 ɛu + u = cos 2 πx 2ɛπ 2 cos 2πx, u(0 = u(1 = 0, with the exact solution u(x = (exp( (1 x/ ɛ + exp( x/ ɛ/(1 + exp( 1/ ɛ cos 2 πx taken from [5]. Techniques which determines the rate of uniform convergence is well-known double mesh principle taken from [5]. In the notation of [5] we have the following tables. Table 1 presents rate of uniform convergence for the scheme (5 as applied to the example (22. Table 1. ɛ/k p ɛ Table 2 shows the difference between the exact and the approximate solution to (22 obtained by (5. Table 2. ɛ/n E E E E E E E E E E E E E E E E E E E E E E E E-12
10 402 M. STOJANOVIĆ In Table 3 is given the rate of the uniform convergence convergence and in Table 4 are given the errors in E = max i u(x i v i norm for the example with non-constant function p(x for modified scheme (5 ɛu + (1 + x 2 u = 4(3x 2 3x + 2(1 + x 2, u(0 = 1, u(1 = 0. Table 3. ɛ/k p ɛ Table 4. ɛ/n E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-06 References [1] A. E. Berger and J. M. Ciment, An Analysis of a Uniformly Accurate Difference Method for a Singular Perturbation Problem, Math. Comput. 37 (1981,
11 SPLINE COLLOCATION METHOD 403 [2] A. Bertozzi and A. Majda, Vorticity and Incompressible Flow, Cambridge Univ. Press, [3] W. Cai and J. Wang, Adaptive Multiresolution Collocation Methods for Initial Boundary Value Problems of Nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996, [4] E. N. Dancer and J. Wei, On the Profile of Solutions with two sharp Layers to a Singularly Perturbed Semilinear Dirichlet Problem, Proceedings of Royal Society of Edinburgh, 127A, (1997, [5] E. P. Doolan, J. J. H. Miller and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, [6] P. A. Farrell, J. J. H. Miller, E. O Riordan and G. I. Shishkin, A Uniformly Convergent Finite Difference Scheme for a Singularly Perturbed Semilinear Equation, SIAM J. Numer. Anal. 33 (1996, [7] J. J. H. Miller and E. O Riordan, The Necessity of Fitted Operators and Shishkin Meshes for Resolving Thin Layer Phenomena, CWI Quarterly 10 (1997, [8] J. J. H. Miller, E. O Riordan and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, Error estimates in Maximum norm of Linear Problems in One and Two Dimension, World Scientific Publishing Co. Pte. Ltd [9] K. W. Morton, Numerical Solution of Convection-Diffusion Problems, Applied Mathematics and Mathematical Computations, Vol. 12, Chapman and Hall, [10] E. O Riordan, Singularly Perturbed Finite Elemet Methods, Numer. Math. 44 (1984, [11] E. O Riordan and M. Stynes, A Uniformly Accurate Finite Element Method for a Singularly Perturbed One-Dimensional Reaction-Diffusion Problem, Math. Comput. 47 (1986, [12] E. O Riordan and M. Stynes, An Analysis of a Super Convergence Result for a Singularly Perturbed Boundary Value Problem, Math. Comput. 46 (1986, [13] H. G. Ross, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion and Flow Problems, Springer Series in Computational Mathematics, Vol. 24, Springer Verlag, [14] M. N. le Roux, Numerical Solution of Nonlinear Reaction Diffusion Processes, SIAM J. Numer. Anal. 37 (2000, [15] M. Stynes and E. O Riordan, Uniformly Accurate Finite Element Method for Singularly Perturbed Problem in Conservative form, SIAM J. Num. Anal. 23 (1986, Institute of Mathematics University of Novi Sad Trg D.Obradovića Novi Sad, Yugoslavia address: stojanovic@im.ns.ac.yu Received: Revised: &
A UNIFORMLY ACCURATE FINITE ELEMENTS METHOD FOR SINGULAR PERTURBATION PROBLEMS. Mirjana Stojanović University of Novi Sad, Serbia and Montenegro
GLASNIK MATEMATIČKI Vol. 38(58(2003, 185 197 A UNIFORMLY ACCURATE FINITE ELEMENTS METHOD FOR SINGULAR PERTURBATION PROBLEMS Mirjana Stojanović University of Novi Sad, Serbia and Montenegro Abstract. We
More informationQuintic B-spline method for numerical solution of fourth order singular perturbation boundary value problems
Stud. Univ. Babeş-Bolyai Math. 63(2018, No. 1, 141 151 DOI: 10.24193/subbmath.2018.1.09 Quintic B-spline method for numerical solution of fourth order singular perturbation boundary value problems Ram
More informationA COMPUTATIONAL TECHNIQUE FOR SOLVING BOUNDARY VALUE PROBLEM WITH TWO SMALL PARAMETERS
Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 30, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu A COMPUTATIONAL
More information1. Introduction We consider the following self-adjoint singularly perturbed boundary value problem: ɛu + p(x)u = q(x), p(x) > 0,
TWMS J. Pure Appl. Math. V., N., 00, pp. 6-5 NON-POLYNOMIAL SPLINE APPROXIMATIONS FOR THE SOLUTION OF SINGULARLY-PERTURBED BOUNDARY VALUE PROBLEMS J. RASHIDINIA, R. MOHAMMADI Abstract. We consider the
More informationNUMERICAL SOLUTION OF GENERAL SINGULAR PERTURBATION BOUNDARY VALUE PROBLEMS BASED ON ADAPTIVE CUBIC SPLINE
TWMS Jour. Pure Appl. Math., V.3, N.1, 2012, pp.11-21 NUMERICAL SOLUTION OF GENERAL SINGULAR PERTURBATION BOUNDARY VALUE PROBLEMS BASED ON ADAPTIVE CUBIC SPLINE R. MOHAMMADI 1 Abstract. We use adaptive
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 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 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 informationA 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 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 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 informationHIGH ORDER VARIABLE MESH EXPONENTIAL FINITE DIFFERENCE METHOD FOR THE NUMERICAL SOLUTION OF THE TWO POINTS BOUNDARY VALUE PROBLEMS
JOURNAL OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4866, ISSN (o) 2303-4947 www.imvibl.org /JOURNALS / JOURNAL Vol. 8(2018), 19-33 DOI: 10.7251/JMVI1801019P Former BULLETIN OF THE
More informationOn the relationship of local projection stabilization to other stabilized methods for one-dimensional advection-diffusion equations
On the relationship of local projection stabilization to other stabilized methods for one-dimensional advection-diffusion equations Lutz Tobiska Institut für Analysis und Numerik Otto-von-Guericke-Universität
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 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 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 informationDRIFT-DIFFUSION SYSTEMS: VARIATIONAL PRINCIPLES AND FIXED POINT MAPS FOR STEADY STATE SEMICONDUCTOR MODELS
DRIFT-DIFFUSION SYSTEMS: VARIATIONAL PRINCIPLES AND FIXED POINT MAPS FOR STEADY STATE SEMICONDUCTOR MODELS Joseph W. Jerome Department of Mathematics Northwestern University Evanston, IL 60208 Abstract
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 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 informationAvailable online at ScienceDirect. Procedia Engineering 127 (2015 )
Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 127 (2015 ) 424 431 International Conference on Computational Heat and Mass Transfer-2015 Exponentially Fitted Initial Value
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 informationUNIFORMLY CONVERGENT 3-TGFEM VS LSFEM FOR SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS ON A SHISHKIN BASED LOGARITHMIC MESH
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 4, Number 4, Pages 35 334 c 23 Institute for Scientific Computing and Information UNIFORMLY CONVERGENT 3-TGFEM VS LSFEM FOR SINGULARLY
More informationRemarks on the blow-up criterion of the 3D Euler equations
Remarks on the blow-up criterion of the 3D Euler equations Dongho Chae Department of Mathematics Sungkyunkwan University Suwon 44-746, Korea e-mail : chae@skku.edu Abstract In this note we prove that the
More informationAn Exponentially Fitted Non Symmetric Finite Difference Method for Singular Perturbation Problems
An Exponentially Fitted Non Symmetric Finite Difference Method for Singular Perturbation Problems GBSL SOUJANYA National Institute of Technology Department of Mathematics Warangal INDIA gbslsoujanya@gmail.com
More informationM.Sc. in Meteorology. Numerical Weather Prediction
M.Sc. in Meteorology UCD Numerical Weather Prediction Prof Peter Lynch Meteorology & Climate Centre School of Mathematical Sciences University College Dublin Second Semester, 2005 2006. In this section
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 informationAn Efficient Algorithm Based on Quadratic Spline Collocation and Finite Difference Methods for Parabolic Partial Differential Equations.
An Efficient Algorithm Based on Quadratic Spline Collocation and Finite Difference Methods for Parabolic Partial Differential Equations by Tong Chen A thesis submitted in conformity with the requirements
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 informationAMS subject classifications. Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50
A SIMPLE FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU AND XIU YE Abstract. The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in
More informationFitted operator finite difference scheme for a class of singularly perturbed differential difference turning point problems exhibiting interior layer
Fitted operator finite difference scheme for a class of singularly perturbed differential difference turning point problems exhibiting interior layer Dr. Pratima Rai Department of mathematics, University
More informationAnalysis of a Streamline-Diffusion Finite Element Method on Bakhvalov-Shishkin Mesh for Singularly Perturbed Problem
Numer. Math. Theor. Meth. Appl. Vol. 10, No. 1, pp. 44-64 doi: 10.408/nmtma.017.y1306 February 017 Analysis of a Streamline-Diffusion Finite Element Method on Bakhvalov-Shishkin Mesh for Singularly Perturbed
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 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 informationCONNECTIONS BETWEEN A CONJECTURE OF SCHIFFER S AND INCOMPRESSIBLE FLUID MECHANICS
CONNECTIONS BETWEEN A CONJECTURE OF SCHIFFER S AND INCOMPRESSIBLE FLUID MECHANICS JAMES P. KELLIHER Abstract. We demonstrate connections that exists between a conjecture of Schiffer s (which is equivalent
More informationA Nodal Spline Collocation Method for the Solution of Cauchy Singular Integral Equations 1
European Society of Computational Methods in Sciences and Engineering (ESCMSE) Journal of Numerical Analysis, Industrial and Applied Mathematics (JNAIAM) vol. 3, no. 3-4, 2008, pp. 211-220 ISSN 1790 8140
More informationAsymptotic analysis of the Carrier-Pearson problem
Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 1,, pp 7 4. http://ejde.math.swt.e or http://ejde.math.unt.e
More informationMemoirs on Differential Equations and Mathematical Physics
Memoirs on Differential Equations and Mathematical Physics Volume 51, 010, 93 108 Said Kouachi and Belgacem Rebiai INVARIANT REGIONS AND THE GLOBAL EXISTENCE FOR REACTION-DIFFUSION SYSTEMS WITH A TRIDIAGONAL
More informationSpline Element Method for Partial Differential Equations
for Partial Differential Equations Department of Mathematical Sciences Northern Illinois University 2009 Multivariate Splines Summer School, Summer 2009 Outline 1 Why multivariate splines for PDEs? Motivation
More informationA Least-Squares Finite Element Approximation for the Compressible Stokes Equations
A Least-Squares Finite Element Approximation for the Compressible Stokes Equations Zhiqiang Cai, 1 Xiu Ye 1 Department of Mathematics, Purdue University, 1395 Mathematical Science Building, West Lafayette,
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 informationASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 11, Number 1, July 004 pp. 189 04 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS Tian Ma Department of
More informationLecture Note III: Least-Squares Method
Lecture Note III: Least-Squares Method Zhiqiang Cai October 4, 004 In this chapter, we shall present least-squares methods for second-order scalar partial differential equations, elastic equations of solids,
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 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 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 informationGlobal regularity of a modified Navier-Stokes equation
Global regularity of a modified Navier-Stokes equation Tobias Grafke, Rainer Grauer and Thomas C. Sideris Institut für Theoretische Physik I, Ruhr-Universität Bochum, Germany Department of Mathematics,
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 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 informationNUMERICAL STUDY OF CONVECTION DIFFUSION PROBLEM IN TWO- DIMENSIONAL SPACE
IJRRAS 5 () November NUMERICAL STUDY OF CONVECTION DIFFUSION PROBLEM IN TWO- DIMENSIONAL SPACE. K. Sharath Babu,* & N. Srinivasacharyulu Research Scholar, Department of Mathematics, National Institute
More information1462. Jacobi pseudo-spectral Galerkin method for second kind Volterra integro-differential equations with a weakly singular kernel
1462. Jacobi pseudo-spectral Galerkin method for second kind Volterra integro-differential equations with a weakly singular kernel Xiaoyong Zhang 1, Junlin Li 2 1 Shanghai Maritime University, Shanghai,
More informationASYMPTOTIC THEORY FOR WEAKLY NON-LINEAR WAVE EQUATIONS IN SEMI-INFINITE DOMAINS
Electronic Journal of Differential Equations, Vol. 004(004), No. 07, pp. 8. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ASYMPTOTIC
More informationPART IV Spectral Methods
PART IV Spectral Methods Additional References: R. Peyret, Spectral methods for incompressible viscous flow, Springer (2002), B. Mercier, An introduction to the numerical analysis of spectral methods,
More information1. Introduction. We consider the model problem that seeks an unknown function u = u(x) satisfying
A SIMPLE FINITE ELEMENT METHOD FOR LINEAR HYPERBOLIC PROBLEMS LIN MU AND XIU YE Abstract. In this paper, we introduce a simple finite element method for solving first order hyperbolic equations with easy
More informationINTRODUCTION TO PDEs
INTRODUCTION TO PDEs In this course we are interested in the numerical approximation of PDEs using finite difference methods (FDM). We will use some simple prototype boundary value problems (BVP) and initial
More informationSingularly Perturbed Partial Differential Equations
WDS'9 Proceedings of Contributed Papers, Part I, 54 59, 29. ISN 978-8-7378--9 MTFYZPRESS Singularly Perturbed Partial Differential Equations J. Lamač Charles University, Faculty of Mathematics and Physics,
More informationarxiv: v1 [math.na] 29 Feb 2016
EFFECTIVE IMPLEMENTATION OF THE WEAK GALERKIN FINITE ELEMENT METHODS FOR THE BIHARMONIC EQUATION LIN MU, JUNPING WANG, AND XIU YE Abstract. arxiv:1602.08817v1 [math.na] 29 Feb 2016 The weak Galerkin (WG)
More informationSolution of a Fourth Order Singularly Perturbed Boundary Value Problem Using Quintic Spline
International Mathematical Forum, Vol. 7, 202, no. 44, 279-290 Solution of a Fourth Order Singularly Perturbed Boundary Value Problem Using Quintic Spline Ghazala Akram and Nadia Amin Department of Mathematics
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 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 informationBlow up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation
Blow up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation Dong Li a,1 a School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 854,
More informationDIRECTION OF VORTICITY AND A REFINED BLOW-UP CRITERION FOR THE NAVIER-STOKES EQUATIONS WITH FRACTIONAL LAPLACIAN
DIRECTION OF VORTICITY AND A REFINED BLOW-UP CRITERION FOR THE NAVIER-STOKES EQUATIONS WITH FRACTIONAL LAPLACIAN KENGO NAKAI Abstract. We give a refined blow-up criterion for solutions of the D Navier-
More informationREGULARITY OF GENERALIZED NAVEIR-STOKES EQUATIONS IN TERMS OF DIRECTION OF THE VELOCITY
Electronic Journal of Differential Equations, Vol. 00(00), No. 05, pp. 5. ISSN: 07-669. UR: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu REGUARITY OF GENERAIZED NAVEIR-STOKES
More informationOn the local existence for an active scalar equation in critical regularity setting
On the local existence for an active scalar equation in critical regularity setting Walter Rusin Department of Mathematics, Oklahoma State University, Stillwater, OK 7478 Fei Wang Department of Mathematics,
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationComputation of operators in wavelet coordinates
Computation of operators in wavelet coordinates Tsogtgerel Gantumur and Rob Stevenson Department of Mathematics Utrecht University Tsogtgerel Gantumur - Computation of operators in wavelet coordinates
More informationRelaxation methods and finite element schemes for the equations of visco-elastodynamics. Chiara Simeoni
Relaxation methods and finite element schemes for the equations of visco-elastodynamics Chiara Simeoni Department of Information Engineering, Computer Science and Mathematics University of L Aquila (Italy)
More informationNumerical methods for the Navier- Stokes equations
Numerical methods for the Navier- Stokes equations Hans Petter Langtangen 1,2 1 Center for Biomedical Computing, Simula Research Laboratory 2 Department of Informatics, University of Oslo Dec 6, 2012 Note:
More informationLocal discontinuous Galerkin methods for elliptic problems
COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2002; 18:69 75 [Version: 2000/03/22 v1.0] Local discontinuous Galerkin methods for elliptic problems P. Castillo 1 B. Cockburn
More informationBlock-Structured Adaptive Mesh Refinement
Block-Structured Adaptive Mesh Refinement Lecture 2 Incompressible Navier-Stokes Equations Fractional Step Scheme 1-D AMR for classical PDE s hyperbolic elliptic parabolic Accuracy considerations Bell
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 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 informationTakens embedding theorem for infinite-dimensional dynamical systems
Takens embedding theorem for infinite-dimensional dynamical systems James C. Robinson Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. E-mail: jcr@maths.warwick.ac.uk Abstract. Takens
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 informationTHE CONVECTION DIFFUSION EQUATION
3 THE CONVECTION DIFFUSION EQUATION We next consider the convection diffusion equation ɛ 2 u + w u = f, (3.) where ɛ>. This equation arises in numerous models of flows and other physical phenomena. The
More informationIterated Defect Correction with B-Splines for a Class of Strongly Nonlinear Two-Point Boundary Value Problems
American Review of Mathematics and Statistics June 2016, Vol. 4, No. 1, pp. 31-44 ISSN: 2374-2348 (Print), 2374-2356 (Online) Copyright The Author(s). All Rights Reserved. Published by American Research
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 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 informationA high order adaptive finite element method for solving nonlinear hyperbolic conservation laws
A high order adaptive finite element method for solving nonlinear hyperbolic conservation laws Zhengfu Xu, Jinchao Xu and Chi-Wang Shu 0th April 010 Abstract In this note, we apply the h-adaptive streamline
More informationNumerical Methods for Two Point Boundary Value Problems
Numerical Methods for Two Point Boundary Value Problems Graeme Fairweather and Ian Gladwell 1 Finite Difference Methods 1.1 Introduction Consider the second order linear two point boundary value problem
More informationNumerical solution of nonlinear sine-gordon equation with local RBF-based finite difference collocation method
Numerical solution of nonlinear sine-gordon equation with local RBF-based finite difference collocation method Y. Azari Keywords: Local RBF-based finite difference (LRBF-FD), Global RBF collocation, sine-gordon
More informationFinite Difference, Finite element and B-Spline Collocation Methods Applied to Two Parameter Singularly Perturbed Boundary Value Problems 1
European Society of Computational Methods in Sciences and Engineering (ESCMSE) Journal of Numerical Analysis, Industrial and Applied Mathematics (JNAIAM) vol. 5, no. 3-4, 2011, pp. 163-180 ISSN 1790 8140
More information1. Introduction. The Stokes problem seeks unknown functions u and p satisfying
A DISCRETE DIVERGENCE FREE WEAK GALERKIN FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU, JUNPING WANG, AND XIU YE Abstract. A discrete divergence free weak Galerkin finite element method is developed
More informationFINITE ELEMENT APPROXIMATION OF ELLIPTIC DIRICHLET OPTIMAL CONTROL PROBLEMS
Numerical Functional Analysis and Optimization, 28(7 8):957 973, 2007 Copyright Taylor & Francis Group, LLC ISSN: 0163-0563 print/1532-2467 online DOI: 10.1080/01630560701493305 FINITE ELEMENT APPROXIMATION
More informationTHE METHOD OF LINES FOR PARABOLIC PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS
JOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS Volume 4, Number 1, Winter 1992 THE METHOD OF LINES FOR PARABOLIC PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS J.-P. KAUTHEN ABSTRACT. We present a method of lines
More informationBoundary Layer Problems and Applications of Spectral Methods
Boundary Layer Problems and Applications of Spectral Methods Sumi Oldman University of Massachusetts Dartmouth CSUMS Summer 2011 August 4, 2011 Abstract Boundary layer problems arise when thin boundary
More informationUNIFORM BOUNDS FOR BESSEL FUNCTIONS
Journal of Applied Analysis Vol. 1, No. 1 (006), pp. 83 91 UNIFORM BOUNDS FOR BESSEL FUNCTIONS I. KRASIKOV Received October 8, 001 and, in revised form, July 6, 004 Abstract. For ν > 1/ and x real we shall
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 informationyou expect to encounter difficulties when trying to solve A x = b? 4. A composite quadrature rule has error associated with it in the following form
Qualifying exam for numerical analysis (Spring 2017) Show your work for full credit. If you are unable to solve some part, attempt the subsequent parts. 1. Consider the following finite difference: f (0)
More informationON THE STRONG SOLUTIONS OF THE INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN
ON THE STRONG SOLUTIONS OF THE INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN XIAN LIAO Abstract. In this work we will show the global existence of the strong solutions of the inhomogeneous
More informationLecture 18 Finite Element Methods (FEM): Functional Spaces and Splines. Songting Luo. Department of Mathematics Iowa State University
Lecture 18 Finite Element Methods (FEM): Functional Spaces and Splines Songting Luo Department of Mathematics Iowa State University MATH 481 Numerical Methods for Differential Equations Draft Songting
More informationNon-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions
Non-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions Wayne McGee and Padmanabhan Seshaiyer Texas Tech University, Mathematics and Statistics (padhu@math.ttu.edu) Summary. In
More informationOn uniqueness in the inverse conductivity problem with local data
On uniqueness in the inverse conductivity problem with local data Victor Isakov June 21, 2006 1 Introduction The inverse condictivity problem with many boundary measurements consists of recovery of conductivity
More informationChapter 5 HIGH ACCURACY CUBIC SPLINE APPROXIMATION FOR TWO DIMENSIONAL QUASI-LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS
Chapter 5 HIGH ACCURACY CUBIC SPLINE APPROXIMATION FOR TWO DIMENSIONAL QUASI-LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS 5.1 Introduction When a physical system depends on more than one variable a general
More informationBOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES
1 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 1.1 Separable Partial Differential Equations 1. Classical PDEs and Boundary-Value Problems 1.3 Heat Equation 1.4 Wave Equation 1.5 Laplace s Equation
More informationSUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS
SUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS A. RÖSCH AND R. SIMON Abstract. An optimal control problem for an elliptic equation
More informationPriority Program 1253
Deutsche Forschungsgemeinschaft Priority Program 1253 Optimization with Partial Differential Equations Klaus Deckelnick and Michael Hinze A note on the approximation of elliptic control problems with bang-bang
More informationA posteriori regularity of the three-dimensional Navier-Stokes equations from numerical computations
A posteriori regularity of the three-dimensional Navier-Stokes equations from numerical computations Sergei I. Chernyshenko, Aeronautics and Astronautics, School of Engineering Sciences, University of
More informationTHE STOKES SYSTEM R.E. SHOWALTER
THE STOKES SYSTEM R.E. SHOWALTER Contents 1. Stokes System 1 Stokes System 2 2. The Weak Solution of the Stokes System 3 3. The Strong Solution 4 4. The Normal Trace 6 5. The Mixed Problem 7 6. The Navier-Stokes
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 information