The H-p Version Of The Finite Element Method For Problems With Nonhomogeneous Dirichlet Boundary Conditions

The finite element method is one of the efficient numerical methods for solving partial dif-ferent equations. According to the spatial structure and the manner of convergence of the finiteelement solution, the finite element method can be divided into three basic forms: h version, pversion and h-p version, in which the h version and the p version are two special cases of the h-pversion.In the framework of the regularity and approximation theory of the Jacobi-weighted Sobolevand Besov spaces, we studied the h-p version of the finite element method for second-order ellipticproblems on polygonal and polyhedral domains with homogeneous or nonhomogeneous Dirichletboundary conditions. The main contribution of this dissertation are as following:1. For the second-order elliptic problems with homogeneous or nonhomogeneous Dirichletboundary conditions in two dimensions, we discussed both cases of the smooth solutions andthe singular solutions on polygonal domains. Under the conditions of the minimum regularityrequirement u∈Hk(?),k > 1, we obtained the optimal H1 norm error estimates of the h-pversion of the finite element method with quasi-uniform meshes.2. For the second-order elliptic problems with homogeneous or nonhomogeneous Dirichletboundary conditions in three dimensions, we discussed the case of smooth solutions. Under theconditions of the minimum regularity requirement u∈Hk(?),k≥1, we obtained the optimal H1norm error estimates of the h-p version of the finite element method with quasi-uniform meshesfor problems with homogeneous Dirichlet boundary conditions. As for the case of problems withnonhomogeneous Dirichlet boundary conditions, we also obtained the optimal H1 norm errorestimates of the h-p version of the finite element method with quasi-uniform meshes under theconditions of u∈Hk(?),k > 32.3. For the second-order elliptic problems with homogeneous Dirichlet boundary conditionson polyhedral domains, the quasi-optimal rate of convergence was obtained for the h-p version ofthe finite element method with quasi-uniform meshes.