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

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