Error estimates for completely discrete FEM in energy-type and weaker norms

TL;DR

This paper develops a unified framework for analyzing error estimates of completely discrete finite element methods applied to boundary value problems with linear diffusion-convection-reaction equations, including stability and optimal error bounds.

Contribution

It introduces a general abstract approach for error analysis of completely discrete FEM without conformity or consistency assumptions, covering stabilized discretizations and various norms.

Findings

Established stability of the discretizations.

Derived optimal error estimates in energy-type norms.

Extended error analysis to weaker norms using Aubin-Nitsche technique.

Abstract

The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion-convection-reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy-type norms and, by generalizing the Aubin-Nitsche technique, optimal error estimates in weaker norms.