Interpolatory super-convergent discontinuous Galerkin methods for nonlinear reaction diffusion equations on three dimensional domains

Abstract

An error analysis of a super-convergent discontinuous Galerkin method formulated in mixed form and applied to a general class of semi-linear equations is presented. To reduce the computational cost at each time step, the nonlinear term is approximated with a Lagrange interpolatory operator. Optimal convergence of order O(hk+1), for both, the primary and auxiliary variables, is obtained for polynomial approximations of degree k. Using a well known element-by-element post-processing procedure, super-convergence of the primary variable is proven. Computational aspects for an efficient implementation are discussed in detail. To avoid solving a large global nonlinear problem at each time step the symmetric Strang operator splitting method is considered as time marching scheme. To validate our error estimates, a series of numerical experiments are carried out using unstructured meshes of three dimensional domains.