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