Design and performance of a space-time virtual element method for the heat equation on prismatic meshes

TL;DR

This paper introduces a space-time virtual element method for solving the heat equation on complex prismatic meshes, demonstrating its convergence and adaptive refinement capabilities for both smooth and singular solutions.

Contribution

It develops a novel virtual element approach for the heat equation on prismatic meshes with variable accuracy, including adaptive mesh refinement strategies.

Findings

Convergence of the method for smooth and singular solutions.

Effective adaptive mesh refinement driven by residual error indicators.

Validation of the method's performance through numerical tests.

Abstract

We present a space-time virtual element method for the discretization of the heat equation, which is defined on general prismatic meshes and variable degrees of accuracy. Strategies to handle efficiently the space-time mesh structure are discussed. We perform convergence tests for the $h$- and $hp$-versions of the method in case of smooth and singular solutions, and test space-time adaptive mesh refinements driven by a residual-type error indicator.