Energy norm error estimates of a hybrid high-order method for the linear parabolic integro-differential equations on general meshes

TL;DR

This paper develops and analyzes a hybrid high-order numerical scheme for linear parabolic integro-differential equations, providing stability and error estimates with validation through numerical experiments.

Contribution

It introduces a novel equal-order hybrid high-order method with proven stability and convergence for these equations on general meshes.

Findings

Error estimates of order O(τ^2 + h^{k+1}) are derived.

Numerical experiments validate the theoretical error bounds.

The method is stable and effective on polygonal meshes.

Abstract

We are concerned in designing a suitable numerical scheme based on the equal-order hybrid high-order (HHO) method for the linear parabolic integro-differential equations. The spatial discretization is made using the equal-order HHO method and subsequently we perform the stability analysis of the corresponding semi-discrete scheme. The convergence results are presented in suitably defined Bochner norms for the semi-discrete problem. Then a second-order temporal discretization is implemented on the time domain using a Crank-Nicolson scheme where the memory term is approximated using a composite trapezoidal quadrature rule. The stability of the resultant complete discrete schemes are analyzed followed by derivation of the error estimates of order $\mathcal{O}(\tau^{2}+h^{k+1})$, $k\ge 0$ is the degree of local polynomial approximation, in discrete $l^{2}(0,T;H^{1}(\Omega))$ and $l^{\infty}(0,T;H^{1}(\Omega))$ like norms. Numerical illustrations are performed on some polygonal meshes validating the theoretical estimates.