We introduce an adaptively weighted Galerkin approach for elliptic problems where diffusion is dominated by strong convection or reaction terms. In such problems, standard Galerkin approximations can have unacceptable oscillatory behavior near boundaries unless the computational mesh is sufficiently fine. Here we show how adaptively weighting the equations within the variational problem can increase accuracy and stability of solutions on under-resolved meshes. Rather than relying on specialized

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... finite elements or meshes, the idea here sets a flexible and robust framework where the metric of the variational formulation is adapted by an approximate solution. We give a general overview of the formulation and an algorithmic structure for choosing weight functions. Numerical examples are presented to illustrate the method. Here, u is the solution, u and ∇u are the Laplacian and gradient of u, ∂ is the boundary of domain , and f is a known data function. We assume that coefficients c and ε are positive constants, that b is a constant vector, and that boundary conditions are homogenous Dirichlet although nonzero boundary data or Neumann/mixed boundary conditions may easily be considered under appropriate smoothness assumptions. When |b| ε, we may consider this as a convectiondominated diffusion problem, which may have solutions with boundary layers downstream from b. When c ε and the reaction term dominates, layer phenomena are also possible. It is well known that standard finite element and finite difference 27 28 YIFEI SUN AND CHAD R. WESTPHAL approaches to such problems can yield solutions with undesirable overshoots and/or oscillatory behavior near these boundary layers when the computational mesh is not sufficiently resolved. The difficulty associated with boundary layers is, of course, not limited to (1-1) but is evident in many applications where diffusive terms are dominated by convective or reactive terms. Throughout this paper, we assume sufficient regularity of the data and domain to ensure solutions are sufficiently smooth, which is a separate issue from boundary layer behavior. There are many well studied numerical approaches to ameliorate layer effects. Through the use of specialized graded meshes [24; 25; 11; 8] or adaptive mesh refinement [9], it is possible to develop a mesh that has sufficient resolution near the layers to resolve the solution and eliminate the effects of the high gradients on the solution in areas where the solution is smooth, which are commonly referred to as the "pollution effects". It is also possible to augment the weak form of the problem by adding mesh-dependent stabilization terms to the formulation [16; 17; 2]. These terms may or may not be consistent with the original problem, but they generally improve the solution on coarse meshes, and their influence diminishes as the mesh is resolved. The variational problem can also be modified through a Petrov-Galerkin formulation, where the test and trial spaces are different. This includes streamline upwind Petrov-Galerkin (SUPG) formulations [5; 6; 18; 1] as well as methods with spaces enhanced by bubble functions [4] . Such problems have also been studied in the context of discontinuous Galerkin (DG) [11; 15] and discontinuous Petrov-Galerkin (DPG) [10; 13] methods. Here continuity requirements in the trial and test spaces are relaxed, and additional degrees of freedom on the element boundaries lead to additional jump conditions in the variational problem. Further comparisons on earlier work for such problems can be found in [22; 12; 14]. Broadly speaking, there are many ingredients in designing a finite element formulation (i.e., reformulating the equations, choosing/adapting the mesh, choosing test/trial spaces, etc.), and improvements on the standard Galerkin approach have been realized by many modifications and combinations of choices in the basic ingredients. In this work, we introduce an adaptively weighted Galerkin finite element approach to (1-1) for cases exhibiting boundary layers. By generalizing the standard Galerkin weak form with weighted inner products, we may essentially redistribute the strength by which the variational problem is enforced across the domain. The use of weighted norms and weighted inner products is, of course, not a new idea. In [21], a weighted Galerkin formulation is used for a parabolic problem where the diffusion coefficient changes sign within the interior of the domain. A weighted Galerkin approach is coupled with a mapping technique in [23] to solve elliptic problems on unbounded domains. And in the least-squares finite element paradigm, using weighted norms to generalize L 2 residual minimization problems allows