High-order interpolatory Serendipity Virtual Element Method for semilinear parabolic problems

TL;DR

This paper introduces an efficient serendipity Virtual Element Method for semilinear parabolic problems on polygonal meshes, reducing computational cost while maintaining high accuracy, validated through theoretical error analysis and numerical experiments.

Contribution

It develops a high-order interpolatory serendipity Virtual Element Method that improves efficiency and accuracy for semilinear parabolic problems on polygonal meshes.

Findings

Reduces degrees of freedom compared to original VEM.

Achieves optimal $L_2$-norm error estimates.

Demonstrates high-order accuracy up to order 6 in numerical tests.

Abstract

We propose an efficient method for the numerical approximation of a general class of two dimensional semilinear parabolic problems on polygonal meshes. The proposed approach takes advantage of the properties of the serendipity version of the Virtual Element Method, which not only reduces the number of degrees of freedom compared to the original Virtual Element Method, but also allows the introduction of an approximation of the nonlinear term that is computable from the degrees of freedom of the discrete solution with a low computational cost, thus significantly improving the efficiency of the method. An error analysis for the semi-discrete formulation is carried out, and an optimal estimate for the error in the $L_2$-norm is obtained. The accuracy and efficiency of the proposed method when combined with a second order Strang operator splitting time discretization is illustrated in our numerical experiments, with approximations up to order $6$.