Continuous Galerkin finite element methods for hyperbolic integro-differential equations

TL;DR

This paper develops and analyzes continuous Galerkin finite element methods for hyperbolic integro-differential equations, establishing well-posedness, regularity, stability, and optimal error estimates for smooth and weakly singular kernels.

Contribution

It introduces a novel finite element approach for hyperbolic integro-differential equations with rigorous stability and error analysis, including duality-based estimates.

Findings

Proved well-posedness and regularity for the model problem.

Established optimal order a priori error estimates.

Demonstrated stability of the discrete dual problem.

Abstract

A hyperbolic integro-differential equation is considered, as a model problem, where the convolution kernel is assumed to be either smooth or no worse than weakly singular. Well-posedness of the problem is studied in the context of semigroup of linear operators, and regularity of any order is proved for smooth kernels. Energy method is used to prove optimal order a priori error estimates for the finite element spatial semidiscrete problem. A continuous space-time finite element method of order one is formulated for the problem. Stability of the discrete dual problem is proved, that is used to obtain optimal order a priori estimates via duality arguments. The theory is illustrated by an example.