Error estimates in balanced norms of finite element methods on Shishkin meshes for reaction-diffusion problems
Provides theoretical justification for using balanced norms in singularly perturbed reaction-diffusion problems, which is important for accurate numerical solutions of boundary layers.
The paper derives error estimates in balanced norms for finite element methods on Shishkin meshes for reaction-diffusion problems, showing that balanced norms correctly reflect layer behavior unlike standard energy norms. The analysis covers anisotropic problems, semilinear equations, supercloseness, and a combination technique.
Error estimates of finite element methods for reaction-diffusion Problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the $H^1$ seminorm leads to a balanced norm which reflects the layer behavior correctly. We discuss also anisotropic problems, semilinear equations, supercloseness and a combination technique.