Convergence Analysis of Virtual Element Methods for the Sobolev Equation with Convection

TL;DR

This paper analyzes the convergence of virtual element methods applied to the Sobolev equation with convection, introducing a new projection operator and providing optimal error estimates validated by numerical experiments.

Contribution

It introduces a novel intermediate projection operator for virtual element methods, enabling optimal convergence analysis for the Sobolev equation with convection.

Findings

Optimal error estimates in energy and L2 norms

Validation of theoretical results through numerical experiments

Assessment of computational efficiency of the proposed methods

Abstract

We explore the potential applications of virtual elements for solving the Sobolev equation with a convective term. A conforming virtual element method is employed for spatial discretization, while an implicit Euler scheme is used to approximate the time derivative. To establish the optimal rate of convergence, a novel intermediate projection operator is introduced. We discuss and analyze both the semi-discrete and fully discrete schemes, deriving optimal error estimates for both the energy norm and L2-norm. Several numerical experiments are conducted to validate the theoretical findings and assess the computational efficiency of the proposed numerical methods.