Error Estimates of a Class of Serendipity Virtual Element Methods for Semilinear Parabolic Integro-Differential Equations on Curved Domains (Journal of Computational Mathematics, 2026)
The rigorous error analysis of a class of serendipity virtual element methods applied to numerically solve semilinear parabolic integro-differential equations on curved domains is the focus of this study. Different from the standard virtual element method, the serendipity virtual element method eliminates all the internal-moment degrees of freedom only under certain conditions of the mesh and the degree of approximation. Consequently, if the interpolation operators are utilized to approximate the nonlinear terms, the implementation of Newton’s iteration algorithm can be simplified. Nonhomogeneous Dirichlet boundary conditions are considered in this paper. The strategy of approximating curved domains with polygonal domains is taken into consideration, and to overcome the issue of suboptimal convergence caused by enforcing Dirichlet boundary conditions strongly, Nitsche-based projection method is employed to impose the boundary conditions weakly. For time discretization, Crank-Nicolson scheme incorporating trapezoidal quadrature rule is adopted. Based on the concrete formulation of Nitsche-based projection method, a Ritz-Volterra projection is introduced and its approximation properties are rigorously analyzed. Building upon these approximation properties, error estimates are derived for the fully discrete scheme. Additionally, the extension of the fully discrete scheme to 3D case is also included. Finally, we present two numerical experiments to corroborate the theoretical findings.
Keywords: Serendipity virtual element method; Curved domain; Nitsche-based projection method; Semilinear parabolic integro-differential equation