Element-wise exponential a posteriori error estimator for singularly perturbed convection-diffusion boundary value problems

Анотація

This paper is devoted to the development of an a posteriori error estimator (AEE) based on exponential basis functions for piecewise-linear finite element approximations on triangular elements. The proposed AEE reliably computes an upper bound for the norm of the approximation error for convection-diffusion boundary value problems. The indicator function of the AEE for each finite element is derived as an analytical solution to an auxiliary boundary value problem, using averaged values of the coefficients of the original differential equation and Dirichlet boundary conditions defined at the nodal values of the computed approximation. Due to its element-wise nature, the computations of the AEE can be efficiently parallelized and adapted for GPGPU computing. The upper error bounds calculated by the proposed AEE in model problems with known exact solutions on uniformly refined meshes demonstrate better efficiency indices compared to a residual-type estimator constructed by polynomial basis function.