P1-Nonconforming Quadrilateral Finite Space with Periodic Boundary Condition and Its Application to Multiscale Problems

Abstract

We consider the P1-nonconforming quadrilateral finite space with periodic boundary condition, and investigate characteristics of the finite space and discrete Laplace operators in the first part of this dissertation. We analyze dimension of the finite element spaces in help of concept of minimally essential discrete boundary conditions. Based on the analysis, we classify functions in a basis for the finite space with periodic boundary condition into two types. And we introduce several Krylov iterative schemes to solve second-order elliptic problems, and compare their solutions. Some of the schemes are based on the Drazin inverse, one of generalized inverse operators, since the periodic nature may derive a singular linear system of equations. An application to the Stokes equations with periodic boundary condition is considered. Lastly, we extend our results for elliptic problems to 3-D case. Some numerical results are provided in our discussion.

In the second part, we introduce a nonconforming heterogeneous multiscale method for multiscale problems. Its formulation is based on the P1-nonconforming quadrilateral finite element, mainly with periodic boundary condition. We analyze a priori error estimates of the proposed scheme by following general framework for the finite element heterogeneous multiscale method. For numerical implementations, we use one of the proposed iterative schemes for singular linear systems in the previous part. Several numerical examples and results are given.