Structure-Preserving Finite Element Method on Topologically Nontrivial Domain

De Rham complexes, their exactness properties, commutative diagram involving them and cohomological technique have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. In this paper we consider topologically nontrivial domains and develop a new structure-preserving finite element method by using the idea of preservation of cohomological space of continuous systems in discretization. Different from the structure-preserving approach on topologically trivial domains developed by D. N. Arnold and R. Hiptmair, this new method not only preserves the commuting diagram between the continuous chain complex and the discrete one on the whole triangulation domain, but also preserves the cohomology space, a crucial topological quantity. We apply the new method to the construction of finite elements for the Dirichlet problem of Poisson equation and elasticity problem, and give some theoretical analysis results. Theoretical analysis and numerical experiments show that the construction of a good finite element method is closely related to the topological structure of domain and the intrinsic property of the system of differential equations.