Regular decomposition and a framework of order reducd methods for fourth order problems

TL;DR

This paper introduces a general framework for reducing the complexity of solving fourth-order problems by decomposing high-regularity spaces into lower-regularity spaces, enabling simpler numerical schemes with optimal error estimates.

Contribution

It presents a novel regular decomposition framework that systematically reduces fourth-order problems to lower-regularity problems, applicable to various equations.

Findings

Framework applicable to multiple fourth-order problems

Discretized schemes achieve optimal error estimates

Demonstrated on three-dimensional problems

Abstract

This paper is devoted to the construction of order reduced method of fourth order problems. A framework is presented such that a problem on a high-regularity space can be deduced in a constructive way to an equivalent problem on three low-regularity spaces which are connected by a regular decomposition, which is corresponding to a decomposition of the figuration of the regularity of the high order space. The framework is fit for various fourth order problems, and the numerical schemes based on the deduced problems can be of lower complicacy. Two fourth order problems in three dimensional are discussed under the framework. They are each corresponding to a regular decomposition, and thus are discretised based on the discretised analogues of the regular decompositions constructed; optimal error estimates are given.