Adaptive Continuous Interior Penalty Finite Element Method And Adaptive Multi-Penalty Discontinuous Galerkin Method

This dissertation addresses the adaptive algorithms of the continuous interior penalty finite element method (CIP-FEM), the continuous multi-penalty finite element method (CMP-FEM) and the multi-penalty discontinuous Galerkin method (MPDG). Convergence and quasi-optimality of these methods are proved.In Chapter 2, we consider the adaptive continuous interior penalty finite element method (ACIP-FEM) for symmetric second order linear elliptic operators. The CIP-FEM uses the same approximation spaces as the finite element method (FEM) but mod-ifies the bilinear form of the FEM by adding a least squares term penalizing the jump of the gradient of the discrete solution at meshes interfaces. This method was successfully applied to convection-dominated problems as a stabilization technique, and recently, it has shown great potential for simulating Helmholtz scattering problems with high wave number. Compared with the analysis for the adaptive finite element method (AFEM) [19], some essential difficulties caused by the introduced penalty term and the restric-tion of H1 regularity of the weak solution need to be treated specially. Finally, under the same conditions as those of AFEM, we prove that when the penalty parameter is not large, the ACIP-FEM is convergent and quasi-optimal. Numerical tests are provided to verify the theoretical results and show advantages of the ACIP-FEM, especially for the Helmholtz equation with high wave number and singularity, the ACIP-FEM performs better than the standard AFEM.In Chapter 3, we consider the adaptive continuous multi-penalty finite elemen-t method (ACMP-FEM), which adds more penalty terms on jumps of higher normal derivatives at mesh interfaces. The additional penalty terms help to reduce efficiently the pollution errors of higher order methods when simulating Helmholtz scatter prob-lems with high wave number. Using the similar tricks as those of ACIP-FEM, we also prove the convergence and quasi-optimality of the ACMP-FEM.In Chapter 4, we consider the adaptive multi-penalty discontinuous Galerkin method (AMPDG). Similar as those for adaptive interior penalty discontinuous Galerkin method (AIPDG) [11], we also prove the convergence and quasi-optimality of the AMPDG. Compared with [11], we introduce the extra penalty terms without any other restrictions for the data in model problem or the regularity of weak solution. Note that when the penalty parameters of the extra penalty terms equal to zero, the AMPDG reduces to the AIPDG, our results extend those of AIPDG [11].In chapter 5, we analyze the adaptive continuous interior penalty finite element method for the Helmholtz problem with high wave number. We prove that when the mesh is fine enough, the ACIP-FEM is convergent. Numerical experiments are provid-ed to verify the theoretical results.