Adaptive mixed virtual element method for the fourth-order singularly perturbed problem

Abstract

The singularly perturbed theory mainly arises in the system of differential equations with the small enough perturbed parameters acting on the highest-order derivatives. In this paper, we introduce the adaptive mixed virtual element method for the fourth-order singularly perturbed problem and the associated eigenvalue problem. It allows to apply the $H^1$-conforming virtual elements to discrete the continuous spaces and reduces the total number of required degrees of freedom of the $H^2$-conforming virtual element method. Basically, the great flexibility of virtual element method becomes appealing in mesh refinement because the locally mesh post-processing to remove hanging nodes is never needed. This naturally motivates us to develop an a posteriori error estimate for the model problem. Based on the numerical solutions, the interior and edge residual terms, and the error terms related to the inconsistency of the virtual element scheme, the error estimators applied to adaptively refine meshes are constructed and then proved to be equivalent to numerical errors under the balanced energy norms. Moreover, we also consider the approximation method for the fourth-order singularly perturbed eigenvalue problem in two-dimensional space. Analogous with the source problem, we not only discuss the boundedness of the eigenfunctions, but also present the upper bound for the error of the approximated eigenvalues by these error estimators. Necessitated by supporting the theoretical analysis, representative numerical examples are reported. We show that the current numerical method converges at the optimal rate uniformly with respect to the singularly perturbed parameters when using the adaptive polygonal meshes.