Morley-Wang-Xu element methods with penalty for a fourth order elliptic singular perturbation problem

Abstract

Two Morley-Wang-Xu element methods with penalty for the fourth order elliptic singular perturbation problem are proposed in this paper, including the interior penalty Morley-Wang-Xu element method and the super penalty Morley-Wang-Xu element method. The key idea in designing these two methods is combining the Morley-Wang-Xu element and penalty formulation for the Laplace operator. Robust a priori error estimates are derived under minimal regularity assumptions on the exact solution by means of some established a posteriori error estimates. Finally, we present some numerical results to demonstrate the theoretical estimates.

Quasi‐uniform convergence analysis of a modified penalty finite element method for nonlinear singularly perturbed bi‐wave problem

“…Furthermore, for ψ ∈ H 2 (Ω), ω h ∈ W h , the following result has been proved in [10], which will be important in the quasi‐uniform convergence analysis. …”

“…However, as we know, there have been a lot of uniform convergence analysis in ϵ about nonconforming FEMs for the following fourth order elliptic singular perturbation problem: where 0 < ϵ ≤ 1 is a real parameter. Such as a C 0 nonconforming FEM [7], the double set parameter method [8, 9], a modified Morley FEM [10], a rectangular Morley element [11], a rectangular Morley FEM on anisotropic meshes [12, 13].…”

“…Although the difference between problem (1) and (2) is subtle, they are fundamentally different equations, as θ is a hyperbolic operator while Δ is an elliptic operator. However, it may be possible to use FEMs for problem (2) in [7–13] as a guide to construct the numerical methods for problem (1). On the other hand, we note that the error estimate in the energy norm with nonconforming FEMs in [3] depended on the negative powers of the perturbation parameter δ , that is, it was divergent when δ → 0.…”

“…where 0 < ≤ 1 is a real parameter. Such as a C 0 nonconforming FEM [7], the double set parameter method [8,9], a modified Morley FEM [10], a rectangular Morley element [11], a rectangular Morley FEM on anisotropic meshes [12,13].…”

Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions

“…In applied sciences, many model problems take the formulation of fourth order elliptic perturbation problems, such as, e.g., the linearized Cahn-Hilliard equation [9,31,39,43,[51][52][53], and the strain gradient problem [1,10,11,28,33,49]. In order for the robust discretisation of such problems, numerical schemes that work for both fourth and second order problems are needed.…”