A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space

TL;DR

This paper develops new error estimates in energy and balanced norms for a time-dependent reaction-diffusion problem with spatial shift, using layer-adapted meshes and a discontinuous Galerkin method, confirmed by numerical results.

Contribution

It introduces the first error estimates in the balanced norm for the problem, employing weighted scalar products and modified bilinear forms.

Findings

Error estimates in energy norm for standard finite element discretization.

Error estimates in balanced norm with modified bilinear form.

Numerical results confirm theoretical predictions and illustrate shift effects.

Abstract

We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Dur\'{a}n- and S-type we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the error of the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Error estimates in the energy norm is for the standard finite element discretization in space, and for the error estimate in the balanced norm the computation of the numerical solution is changed by using a different bilinear form. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution.