Among domains of particular interest in applied sciences are polygons, potentially including non-convex corners. The presence of corners in the computational domains negatively influences the regularity of the solutions of elliptic partial differential equations, by introducing a particular known type of singular functions into the solution. The appearance of singular functions in the solutions on polygonal domains diminishes the accuracy of the standard finite element method. Furthermore, the approximation is suboptimal not only in the vicinity of the corner but also when considered in some positive distance from it - the behaviour known as the pollution effect.

Many approaches for improving the global convergence of the finite element solution exist already, such as adaptivity, graded mesh algorithms or enrichment of the finite element space by singular functions. These methods improve the approximation quality regaining the optimal convergence properties. However, they require a modification of the standard finite element solution in a neighbourhood of the corners and require changing the structure of the mesh or the function space, in which the solution is approximated.

We propose an alternative approach - the so-called energy-correction, which is based on the observation that the approximation quality of the finite element solution when measured in the L2(Ω)-norm, is limited by the approximation of the energy. The numerical algorithm modifies the bilinear form governing the system of equations so that the optimal approximation of the energy is regained. Using the energy-correction method, optimal convergence concerning the interpolation error is regained. As opposed to the schemes involving adaptivity or the mesh grading, the energy-correction works on uniformly refined meshes. Since the method was first introduced for the scalar elliptic equations, it has been extended to the higher-order elements, parabolic problem, fluid dynamics and optimal control problems.