Supraconvergence of a Non-Uniform Discretisation for an Elliptic Third-Kind Boundary-Value Problem with Mixed Derivatives

Transcription

1 Supraconvergence of a Non-Uniform Discretisation for an Elliptic Third-Kind Boundary-Value Problem with Mixed Derivatives Etienne Emmrich Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 1063 Berlin, Germany emmrich@math.tu-berlin.de Abstract. The third-kind boundary-value problem for an elliptic equation of second order with variable coefficients and mixed as well as firstorder terms on a domain that is the union of rectangles is approximated by a linear finite element method with first-order accurate quadrature. The scheme is equivalent to a standard finite difference method. Although the discretisation is in general only first-order consistent, supraconvergence, i.e. convergence of higher order, is shown to take place even on non-uniform grids. Local error estimates of optimal order mins, 3/ in the H 1 Ω-norm can be derived for s [1, ] if the exact solution is in the Sobolev-Slobodetskij space H 1+s Ω. If no mixed derivatives occur then optimal order s can be achieved. The supraconvergence also implies the supercloseness of the gradient. Key words: elliptic PDE, fully discrete FEM, FDM, non-uniform grid, supraconvergence, supercloseness of gradient MSC 000: 65N1; 65N15; 65N06; 65N30 1 Introduction We consider the discretisation of the differential equation Au := au x x bu x y bu y x cu y y +du x +eu y +fu = g in Ω IR 1 with variable coefficients subject to Robin boundary conditions Bu := au x η x + bu x η y + bu y η x + cu y η y + αu = ψ on Γ := Ω, where the domain Ω is the union of rectangles and η x, η y denotes the outer normal on Γ. The coefficients in A and B as well as ψ are assumed to be sufficiently smooth such that all are at least uniformly continuous and g L Ω. Moreover, A is assumed to be uniformly elliptic and the corresponding homogeneous problem is supposed to be uniquely solvable. The corresponding bilinear form needs not to be strongly positive.

2 The discretisation is obtained from linear finite elements on a triangulation T H of Ω, which relies upon a non-uniform rectangular grid Ω H, in combination with an appropriate quadrature that is of first order. Although the scheme is in general only first-order consistent, higher-order convergence can be proven: Let u H denote the discrete solution on Ω H, P H the piecewise linear interpolation with respect to T H note that P H u H is the finite element solution, R H the pointwise restriction on Ω H, and 1, the usual H 1 Ω-norm. For a quasi-uniform sequence of grids Ω H with a maximum mesh-size H max tending to zero, we derive a local error estimate from which the global error estimate P H R H u P H u H 1, = OHmax mins,3/ for s [1, ] if u H 1+s Ω follows. In the case of no mixed derivatives, optimal order OHmax s can be proven. For s {1, }, the estimates even hold without the assumption of quasi-uniformity. As a consequence, a global estimate of order OH max s+1/ in the case of mixed derivatives and of order OHmax s if no mixed derivatives occur is obtained for s [1, ] even for an arbitrary sequence of non-uniform grids. Note that these supraconvergence results also show the supercloseness property for the gradient in the context of finite elements. Convergence results of order OHmax s for s 1/, 1 can also be obtained but are not included here. Super- and supraconvergence of finite element and finite difference solutions have been considered by many authors; we refer to [1,, 7 1] and the references cited therein. In particular, we continue the work presented in [4 6]. For the third-kind boundary-value problem 1, on a domain Ω that is the union of rectangles but without mixed or first-order derivatives b = d = e = 0, local error estimates showing supraconvergence of order s for s 1/, ] if u H 1+s Ω are proved in [4]. The aim of this paper is to present a generalisation of the results of [4] that allows mixed and first-order derivatives in the differential operator A. An essential step in the analysis is the equivalence of the fully discrete finite element method with quadrature to a finite difference method. Discretisation The corresponding variational problem reads as find u H 1 Ω such that Au, v = g, v + ψ, v Γ for all v H 1 Ω with the sesquilinear form Av, w := av x, w x + bv x, w y + bv y, w x + cv y, w y + dv x, w + ev y, w + fv, w + αv, w Γ, v, w H 1 Ω. Here and in the following, we employ the usual notation for Lebesgue-, Sobolev-, Sobolev-Slobodetskij spaces and spaces of continuously differentiable functions. In particular, we denote by, and, Γ the inner product on L Ω and L Γ, respectively, and by r,p,d the usual norm on W r,p D for a domain D where we omit the subscript D if D = Ω. As the boundary Γ is only

3 Lipschitz, the norm of Sobolev-Slobodetskij spaces on Γ shall be defined through summing up over disjoint straight boundary sections. For the discretisation, let h = {h j } j ZZ and k = {k l} l ZZ be two sequences of positive real numbers and consider the two-dimensional grid IR H := IR x h IR y k, where IR x h := {x j IR : x j+1 := x j + h j, j ZZ} for x 0 IR given; IR y k is defined analogously. Moreover, let x j+1/ := x j +h j / = x j+1 h j / := x j+1 1/ with an analogous notation in the y-direction. We define Ω H := Ω IR H, Γ H := Γ IR H, Ω H := Ω IR H = Ω H Γ H and consider a sequence {Ω H } H Λ of grids with H max := max{h j, k l : j, l ZZ} tending to zero. We say that {Ω H } H Λ is quasi-uniform if all possible quotients of mesh sizes of Ω H are bounded independently of H. The vertices of Ω are assumed to be in Γ H. The triangulation T H is supposed to be a set of open triangles in which the vertices are the grid points of Ω H. Throughout this paper, we assume H max being sufficiently small. By W H, we denote the space of grid functions on Ω H. For convenience, we tacitly assume that a function v H W H is extended on IR H by zero. We often write v P instead of v H P. For P = x j, y l Ω H, let Then v H, w H H := P := x j 1/, x j+1/ y l 1/, y l+1/ Ω, Γ P := x j 1/, x j+1/ y l 1/, y l+1/ Γ. P Ω H P v P w P and φ H, χ H Γ,H := Γ P φ P χ P defines an inner product on W H and on the space of grid functions on Γ H, respectively. Let A H := a H + b H + c H + d H + e H + f H + α H be the sesquilinear form defined by a H v H, w H := P H v H x P H w H x dv, T H a,x d H v H, w H := dp H w H,x P H v H x dv, T H f H v H, w H := R H fv H, w H H, α H v H, w H := R H αv H, w H Γ,H, and with forms c H and e H defined analogously to a H and d H, respectively. Here, the subscript, x denotes the value at the midpoint of the side of T H parallel to the x-axis. The definition of the form b H is based upon two special triangulations T 1 H and T H.

4 Let j,l denote an open triangle having an angle π/ at x j, y l IR H and two adjacent grid points as further vertices. We then define { } T ν H := j,l Ω : x j, y l IR H with j + l + ν being odd, ν = 1,, and associate the piecewise linear interpolation P ν H. Then b H v H, w H := 1 b 1 H v H, w H + b H v H, w H, where for ν = 1, b ν H v H, w H := T ν H b P ν H v H x P ν H w H y + P ν H v H y P ν H w H x dv with b being the value of b at the vertex of T ν H that corresponds with the angle π/. The fully discrete Galerkin approximation now reads as find u H W H such that A H u H, v H = g H, v H H + ψ H, v H Γ,H for all v H W H 3 with right-hand side g H and boundary value ψ H given by g P := P 1 P gdv P Ω H, ψ P := ψp P Γ H. 4 It can be shown that 3 is equivalent to the finite difference approximation A H u H := δ x 1/ aδ x 1/ u H δ y bδ x u H δ x bδ y u H δ y 1/ cδ y 1/ u H + R H dδ x u H + R H e δ y u H + R H f u H = g H in Ω H, supplemented by an appropriate approximation of the boundary condition see also [4]. In particular, we have A H v H, w H = A H v H, w H H for all v H, w H W H with w H = 0 on Γ H. Here, we use the divided differences δ x 1/ v j,l := v j+1/,l v j 1/,l, δ x 1/ x j+1/ x j 1/ δ x v j,l := v j+1,l v j 1,l x j+1 x j 1 v j+1/,l := v j+1,l v j,l x j+1 x j, and corresponding differences in the y-direction. The following stability result can be proven similarly as [5, Thm. ].

5 Proposition 1. For Ω H H Λ and v H W H, the following estimate holds true: A H v H, w H P H v H 1, C sup. 0 w H W H P H w H 1, By C, we denote a generic constant that is independent of significant quantities such as the grid size. Proposition 1 implies the unique solvability of the discrete problem. 3 Supraconvergence of the Discretisation For P = x j, y l Γ y H, the set of grid points lying on Γ y which is the part of Γ that is parallel to the y-axis, we define Γ P := {x j } y l 1/, y l+1/ Γ. Moreover, we use the convention that, depending on the location of the domain Ω, η x and η y always take the value +1 or 1 on the sections of the boundary Γ even at the vertices of the domain. The main result reads as Theorem 1. Let u H Ω, a, b, c, d, e, f W 1, Ω, α W 1, Γ, ψ H 1 Γ if s = 1 and let u H 1+s Ω, a, b, c, d, e, f W,/ s Ω, α W,1/ s Γ, ψ H s Γ if s 1, ]. Moreover, assume that {Ω H } H Λ is quasi-uniform if s {1, }. The discretisation error then satisfies the estimate P H R H u P H u H 1, C diam P s u 1+s,, P P Ω H + CH mins,3/ max in the case of mixed derivatives. If b 0 then P H R H u P H u H 1, C ΓP mins,3/ u 1/+s,,Γ P + Γ P s ψ s,,γ P 1/ + u 1+s, + u 1/+s,,Γ + ψ s,,γ P Ω H diam P s u 1+s,, P Γ P s u s,,γp + ψ s,,γ P 1/ 5 CH s max u 1+s, + u s,,γ + ψ s,,γ. 6 Sketch of Proof In what follows, we sketch the main steps in the proof of Theorem 1 focussing only on the main part of the differential operator as the lower-order terms can be dealt with somewhat simpler. For the discretisation error, we find from Proposition 1 τ H w H P H R H u P H u H 1, C sup 0 w H W H P H w H 1,

6 with the truncation error τ H w H := A H R H u, w H A H u H, w H where = τ a H w H + τ b H w H + τ c H w H + τ d H w H + τ e H w H + τ f H w H, τ a H w H := a H R H u, w H + P Ω H P au x x dv w P Γ P au x η x Pw P and the other parts of τ H corresponding to b H,...,f H are defined analogously. We commence with the case s = 1. The estimate of τ a H w H relies upon the decomposition τ a H w H = τ a H,1 w H + τ a τ a H, w H := H, w H with au x η x dσ Γ P au x η x P w P. Γ P Analogous decompositions are at hand for τ b H w H and τ c H w H. The terms τ a H,1 w H and τ c H,1 w H satisfy the estimate desired as is shown in [4, Sect. 4]. Also τ b H,1 w H, the truncation error related to mixed derivatives, can be estimated as desired as is shown in the following. The proof starts with an integration and summation by parts, rewriting τ b H,1 w H in terms of contributions on rectangles j+1/,l+1/ := x j, x j+1 y l, y l+1 Ω, and an application of the Bramble-Hilbert lemma. We find see also [4, Lemmata 4.7, 4.8] τ a H, w H + τ b H, w H + τ c H, w H = ψ αudσ Γ P ψ αup w P P Γ Γ P H 1/ C P H w H 1,, Γ P u 1,,Γ P + ψ 1,,Γ P which finally proves the result for s = 1. Note that u 1,,ΓP u 3/,,ΓP. We come to the case s 1, ]. For P Γ y 1/, the set of points x j, y l+1/ lying on parts of Γ parallel to the y-axis, let P := x j, y l, P + := x j, y l+1, and Γ P := {x j } y l, y l+1, Γ P := {x j} y l, y l+1/, Γ + P := {x j} y l+1/, y l+1. The estimates are based upon the decomposition τ a H w H = τ a H,3 w H + τ a H,4 w H with

7 τ a H,4 w H := τ a H, w H 1 Γ P P Γ y 1/ Γ P Γ + P au x η x dσ + Γ P au xη x P au x η x dσ Γ P au xη x P + δ y 1/ w P and analogous decompositions for τ b H w H and τ c H w H. The terms τ a H,3 w H and τ c H,3 w H satisfy local estimates of order s see [4, Sect. 5] and it remains to consider τ b H,3 w H. With integration and summation by parts, τ b H,3 w H can again be rewritten in terms of contributions on rectangles j+1/,l+1/. What follows, is a rather intrigued decomposition and multiple application of the generalised Bramble-Hilbert see [3] and bilinear lemma. After all, the boundary contribution Γ P η x P bp + u y P + u y P + bp u y P u y P 4 P Γ y 1/ δ 1/ y w P 7 plus an analogous contribution from parts of Γ parallel to the x-axis has to be estimated. Unfortunately, with the aid of the generalised Bramble-Hilbert lemma, we can only derive an estimate of maximum order 3/ in terms of u 1/+s,,Γ. Here, we also employ the continuous embedding H 1 Ω H 1/ Γ and an identification of P H w H 1/,,Γ in terms of differences of values of w at the boundary Γ. Some straightforward calculations finally show that τ a H,4 w H + τ b H,4 w H + τ c H,4 w H = Γ P ψ αudσ ψ αup + + ψ αup w P ++ w P + Γ P Γ P ψ αudσ ψ αup + + ψ αup w P ++ w P. Γ P P Γ x 1/ P Γ y 1/ We are thus left with trapezoidal rules that lead to the estimate of order s see also [4, Lemma 5.8] and note that u s,,γp C u 1/+s,,ΓP. Here, it is important to have the assumption ψ H s Γ. By interpolation using the estimates 5, 6 of Theorem 1 for s = 1 and s =, which is then valid without the assumption of quasi-uniformity, the following corollary is derived. Corollary 1. Let s [1, ] and assume that u H 1+s Ω, a, b, c, d, e, f W, Ω, α W, Γ, and ψ H s Γ. The discretisation error then satisfies the estimate P H R H u P H u H 1, CH 1+s/ max u 1+s, + u 1/+s,,Γ + ψ s,,γ

8 in the case of mixed derivatives. If b 0 then P H R H u P H u H 1, CH s max u 1+s, + u s,,γ + ψ s,,γ. As is shown in [4, Remark 5.3, 5.4], the averaged restriction of the right-hand side g in 4 can be replaced by the pointwise restriction on Ω H if g H s Ω for s 1, ] retaining the order of convergence. If, however, g is not smooth enough then the pointwise restriction may destroy the higher order. References 1. D. Bojović and B.S. Jovanović, Fractional order convergence rate estimates of finite difference method on nonuniform meshes, CMAM J.H. Brandts, Superconvergence phenomena in finite element methods Proefschrift, Universiteit Utrecht, T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp E. Emmrich and R.D. Grigorieff, Supraconvergence of a finite difference scheme for elliptic third kind boundary value problems in fractional order Sobolev spaces, CMAM J.A. Ferreira and R.D. Grigorieff, On the supraconvergence of elliptic finite difference schemes, Appl. Numer. Math J.A. Ferreira and R.D. Grigorieff, Supraconvergence and supercloseness of a scheme for elliptic equations on non-uniform grids, Numer. Funct. Anal. Optimiz B.S. Jovanović, Finite difference schemes for partial differential equations with weak solutions and irregular coefficients, CMAM B.S. Jovanović, The finite difference method for boundary-value problems with weak solutions Posebna Izdanja, 16, Matematički Institut u Beogradu, M. Křížek and P. Neittaanmäki, Bibliography on superconvergence, in: M. Křížek, P. Neittaanmäki, and R. Stenberg, eds., Finite element methods Marcel Dekker, New York, L.B. Wahlbin, Superconvergence in Galerkin finite element methods, Lect. Notes in Math Springer, Berlin, Q. Zhu, A survey of superconvergence techniques in finite element methods, in: M. Křížek, P. Neittaanmäki, and R. Stenberg, eds., Finite element methods Marcel Dekker, New York, A. Zlotnik, On superonvergence of a gradient for finite element methods for an elliptic equation with the nonsmooth right-hand side, CMAM