Optimal error estimates for non-conforming approximations of linear parabolic problems with minimal regularity

TL;DR

This paper derives optimal error estimates for non-conforming finite element approximations of linear parabolic problems, applicable even with minimal regularity of the solution, using a unified inf-sup approach.

Contribution

It provides the first error estimate for non-conforming methods that does not require additional regularity assumptions on the solution.

Findings

Error estimate matches the sum of interpolation and conformity errors.

The approach applies to various methods including conforming, nonconforming, and discontinuous Galerkin.

Numerical examples confirm the theoretical error bounds.

Abstract

We consider a general linear parabolic problem with extended time boundary conditions (including initial value problems and periodic ones), and approximate it by the implicit Euler scheme in time and the Gradient Discretisation method in space; the latter is in fact a class of methods that includes conforming and nonconforming finite elements, discontinuous Galerkin methods and several others. The main result is an error estimate which holds without supplementary regularity hypothesis on the solution. This result states that the approximation error has the same order as the sum of the interpolation error and the conformity error. The proof of this result relies on an inf-sup inequality in Hilbert spaces which can be used both in the continuous and the discrete frameworks. The error estimate result is illustrated by numerical examples with low regularity of the solution.