Decoupled mixed element schemes for fourth order problems

TL;DR

This paper introduces a general decoupling process for mixed element schemes that simplifies the discretization of fourth order problems, especially applied to 3D bi-Laplacian equations, using low-order finite elements.

Contribution

It presents a novel, general decoupling method transforming high-regularity elliptic problems into low-order systems for efficient discretization.

Findings

Effective decoupling of fourth order problems into low-order systems

Application to 3D bi-Laplacian equations with finite elements

Flexible framework for various fourth order problems

Abstract

In this paper, we study decoupled mixed element schemes for fourth order problems. A general process is designed such that an elliptic problem on high-regularity space is transformed to a decoupled system with spaces of low order involved only and is further discretised by low-degree finite elements. The process can be fit for various fourth order problems, and is used in the remaining of the paper particularly for three-dimensional bi-Laplacian equation to conduct a family of mixed element discretisation schemes.