Constructive error analysis of a full-discrete finite element method for the heat equation

TL;DR

This paper introduces a new, easy-to-implement finite element method for the heat equation, providing constructive error estimates and verified stability, extending previous work with a focus on practical computational guarantees.

Contribution

It presents a novel full-discrete finite element scheme for the heat equation with constructive error analysis and verified stability, simplifying implementation compared to previous methods.

Findings

The method is numerically stable with verified computations.

Constructive error estimates are effectively used for guaranteed accuracy.

The scheme is comparable to a standard Galerkin method and easier to implement.

Abstract

In this paper, we present a new full-discrete finite element method for the heat equation, and show the numerical stability of the method by verified computations. Since, in the error analysis, we use the constructive error estimates proposed ny Nakao et. all in 2013, this work is considered as an extention of that paper. We emphasize that concerned scheme seems to be a quite normal Galerkin method and easy to implement for evolutionary equations comparing with previous one. In the constructive error estimates, we effectively use the numerical computations with guaranteed accuracy.