Error Analysis of Serendipity Virtual Element Methods for Semilinear Parabolic Integro-Differential Equations

The main objective of this study is to evaluate the performance of serendipity virtual element methods in solving semilinear parabolic integro-differential equations with variable coefficients. The primary advantage of this method, in comparison to the standard (enhanced) virtual element methods, lies in the reduction of internal-moment degrees of freedom, which can speed up the iterative algorithms when using the quasi-interpolation operators to approximate nonlinear terms. The temporal discretization is obtained with the backward-Euler scheme. To maintain consistency with the accuracy order of the backward-Euler scheme, the integral term is approximated using the left rectangular quadrature rule. Within the serendipity virtual element framework, we introduced a Ritz–Volterra projection and conducted a comprehensive analysis of its approximation properties. Building upon this analysis, we ultimately provided optimal \(H^1\)-seminorm and \(L^2\)-norm error estimates for both the semi-discrete and fully discrete schemes. Finally, two numerical examples that serve to illustrate and validate the theoretical findings are presented.