Error Estimations In The Balanced Norm Of Finite Element Method On Bakhvalov-Shishkin Triangular Mesh For Reaction-Diffusion Problems

A balanced norm, rather than the common energy norm, is introduced to reflect the behavior of layers more accurately in the finite element method for singularly perturbed reaction-diffusion problems. Convergence of optimal order in the balanced norm has been proved in the case of rectangular finite elements. However, for triangular finite elements P-k (k >= 2), it is still open to prove this convergence result. With the help of the L-2-stability of a weighted L-2 projection, instead of the L-infinity-stability widely used in existing references, the geometric constraints on meshes are relaxed. As a result, the optimal order convergence in the balanced norm is proved in the case of Bakhvalov-Shishkin triangular meshes. Numerical experiments support theoretical results. (C) 2021 Elsevier Ltd. All rights reserved.