Computer Methods in Applied Mechanics and Engineering
Analysis of a new stabilized higher order finite element method for advection–diffusion equations
Introduction
We consider the two-point boundary value problemwith sufficiently smooth functions b, c, f, and a small positive parameter 0 < ε ≪ 1. We assume thatwhich guarantees the unique solvability of the problem.
Standard Galerkin-type finite element methods exhibit spurious oscillations unless the mesh is very fine. Therefore, a number of stabilized methods (SUPG, Galerkin-least squares, residual free bubble, etc.) have been developed and extended both to the multi-dimensional case and to the incompressible Navier–Stokes equations; see [4] for the beginning and [15], [16] for a survey. The variational multiscale method [8], [9] has been introduced as a framework for a better understanding of the fine-to-coarse scale effects and as a platform for the development of new numerical methods. Recently, Hughes and Sangalli [10], [11] succeeded in giving explicit formulas for the fine-scale Green’s function arising in variational multiscale analysis. An important observation of their approach is that the fine-scale problem for higher order finite element approximations can be considered as a constrained bubble problem.
For the constant coefficient case (with c = 0) and piecewise linear finite elements the close relations between the SUPG method, the residual free bubble approach, and the variational multiscale method are well-known. However, when using higher order finite elements the variational multiscale approach leads to a new stabilized method which seems to be not analyzed up to now. In order to distinguish this new method from the SUPG method we will call it variational multiscale (VMS) method.
The main objective of the paper is to analyze the VMS method and to give error estimates in several norms on different types of meshes. In Section 2 we shortly describe the variational multiscale approach leading to the new numerical method and show an alternative way for its derivation. Then, in Section 3 we give an error estimate in a mesh dependent norm which is related to the discrete bilinear form. It is important to note that in the case of higher order finite elements a new interpolation has been used to get the desired error estimates. Section 4 is devoted to ε-uniform error estimates on families of Shishkin meshes. These estimates are based on a decomposition of the solution into a smooth and layer part, respectively, as well as a detailed study of approximation properties in and outside the layer region. It turns out that the error of the new VMS method to the interpolant is of order k + 1/2 uniformly in ε for piecewise polynomials of degree k. Using superconvergence properties in the case k = 1 the accuracy can be enhanced to almost second order. Finally, we show that by a proper postprocessing the same bounds can be established for the error of the postprocessed numerical solution to the solution itself.
Notations. Throughout the paper C will denote a generic positive constant that is independent of ε and the mesh.
We use the standard Sobolev spaces Wk,p(D), Hk(D) = Wk,2(D), , Lp(D) = W0,p(D) for nonnegative integers k and 1 ⩽ p ⩽ ∞ and write (·, ·)D for the L2(D) inner product. Here D is any measurable subset of (0, 1). Then, ∣·∣k,p,D and ∥·∥k,p,D are the usual Sobolev seminorm and norm on Wk,p(D). When D = (0, 1) we drop D from the notation for simplicity. We will also simplify the notation in the case p = 2 by setting ∥·∥k,D = ∥·∥k,2,D and ∣·∣k,D = ∣·∣k,2,D.
Section snippets
Variational multiscale method
The weak formulation of (1) is given by
Find such that for all v ∈ VThe idea of the variational multiscale approach is to split the solution space V into resolvable and unresolvable scales. This is realized by choosing a finite element space Vh which represents the resolvable scales and a projection operator P : V → Vh such thatNow the weak formulation (3) can be reformulated as
Find uh ∈ Vh and such that
Error analysis on an arbitrary mesh
In this section we want to study the convergence properties of the method on an arbitrary mesh. The smoothness of c and the application of inverse inequalities guarantee the existence of a general constant cmax such thatThe constant cmax depends on the polynomial degree k, but to simplify the notation we will not indicate this. In the following, we assume that the user chosen VMS parameter τK satisfiesNote that
Decomposition of the solution
First, we summarize some analytical properties of the two-point boundary value problemwith sufficiently smooth functions b, c, f, and a small positive parameter 0 < ε ≪ 1. We assume thatsuch that a boundary layer of exponential type appears in the neighbourhood of x = 1 and internal layers are excluded. It is well-known that the solution can be decomposed intowhere for l = 0, … , L and x ∈ [0, 1]
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