Quasi-optimality of adaptive finite element methods for controlling local energy errors

Abstract

A rich theory demonstrating convergence and optimality of adaptive finite element methods (AFEM) has been developed in recent years. In this work we prove optimality of AFEM which are designed to control local energy errors in elliptic partial differential equations. Because errors propagate globally in FEM, controlling local errors requires controlling both local energy solution properties and global error contributions (pollution errors) which may be measured in a weaker norm such as the $$L_2$$L2 norm. We define adaptive methods which control both of these error components and prove that they converge with the best possible rate over all possible refinements of the initial mesh. These results are valid for Poisson's problem on convex polyhedral domains in arbitrary space dimension. Our theory establishes AFEM optimality for several adaptive marking strategies which rigorously control pollution effects. We also present numerical examples that illustrate our theory and confirm that local energy AFEM without pollution control can fail to yield optimal meshes.