The AL basis for the solution of elliptic problems in heterogeneous media (Q2909536)

The authors consider an elliptic Poisson-type problem with sufficiently smooth diffusion coefficients, right-hand side, and domain boundary. Then a discretization with continuous piecewise linear finite elements on a finite element mesh with mesh width \(H\) converges with respect to the energy norm at a rate of \(O(H)\). If the diffusion coefficient is nonsmooth as it is typical for heterogeneous media, the convergence rate becomes very poor. The authors address the question of whether the linear finite element space can be enriched by ``a few'' additional shape functions so that the convergence rate is \(O(H)\) without any regularity assumption on the diffusion coefficient. The proposed approach requires only the solution of the partial differential equation for a localized right-hand side (which can be efficiently performed, e.g., with an algebraic multigrid algorithm) and, as a generalized finite element space, it can be applied to any right-hand side \(f \in L^2 (\Omega)\) without any modification. This approach guarantees that the AL basis converges with a rate of \(O(H)\).