Numerical solution of parabolic problems based on a weak space-time formulation

TL;DR

This paper develops a novel weak space-time formulation for the heat equation that retains a pointwise solution component, leading to a quasi-optimal and pointwise superconvergent numerical scheme with proven error estimates.

Contribution

It introduces a new formulation that includes the pointwise component, enabling improved numerical accuracy and convergence for parabolic problems.

Findings

The scheme is quasi-optimal in the L2 sense.

The scheme exhibits pointwise superconvergence at temporal nodes.

Numerical experiments confirm theoretical error estimates.

Abstract

We investigate a weak space-time formulation of the heat equation and its use for the construction of a numerical scheme. The formulation is based on a known weak space-time formulation, with the difference that a pointwise component of the solution, which in other works is usually neglected, is now kept. We investigate the role of such a component by first using it to obtain a pointwise bound on the solution and then deploying it to construct a numerical scheme. The scheme obtained, besides being quasi-optimal in the $L^2$ sense, is also pointwise superconvergent in the temporal nodes. We prove a priori error estimates and we present numerical experiments to empirically support our findings.