A priori and a posteriori error analysis of the lowest-order NCVEM for second-order linear indefinite elliptic problems

TL;DR

This paper analyzes the nonconforming virtual element method for second-order indefinite elliptic PDEs, providing both a priori and a posteriori error estimates, and demonstrates improved adaptive mesh refinement through numerical experiments.

Contribution

It introduces a new error analysis framework for NCVEM applied to indefinite elliptic problems, including stability, existence, and optimal error estimates.

Findings

Optimal error estimates in $H^1$ and $L^2$ norms.

Reliable and efficient residual-based a posteriori error estimator.

Adaptive mesh refinement improves convergence rates.

Abstract

The nonconforming virtual element method (NCVEM) for the approximation of the weak solution to a general linear second-order non-selfadjoint indefinite elliptic PDE in a polygonal domain is analyzed under reduced elliptic regularity. The main tool in the a priori error analysis is the connection between the nonconforming virtual element space and the Sobolev space $H^1_0(\Omega)$ by a right-inverse $J$ of the interpolation operator $I_h$. The stability of the discrete solution allows for the proof of existence of a unique discrete solution, of a discrete inf-sup estimate and, consequently, for optimal error estimates in the $H^1$ and $L^2$ norms. The explicit residual-based a posteriori error estimate for the NCVEM is reliable and efficient up to the stabilization and oscillation terms. Numerical experiments on different types of polygonal meshes illustrate the robustness of an error estimator and support the improved convergence rate of an adaptive mesh-refinement in comparison to the uniform mesh-refinement.