QUASI-OPTIMAL NONCONFORMING METHODS FOR SYMMETRIC ELLIPTIC PROBLEMS

In this PhD thesis we characterize quasi-optimal nonconforming methods for symmetric elliptic linear variational problems and investigate their structure. The abstract analysis is complemented by various applications and numerical tests in the finite element framework. In the first part of the thesis we introduce a rather large class of nonconforming methods, mimicking the variational structure of the model problem. Then, we characterize the subclass of quasi-optimal methods in terms of suitable notions of stability and consistency. We determine also the quasi-optimality constant and observe its dependence on the proposed notions of stability and consistency. For this purpose, we introduce an appropriate stability constant and two consistency measures. The second part of the thesis is devoted to exemplify the application of the above-mentioned results through the construction of various quasi-optimal nonconforming finite element methods. We consider the following three model problems: the Poisson problem, the linear elasticity problem and the biharmonic problem. For each one of them, we propose approximation methods based on discontinuous elements and/or classical nonconforming elements, such as the Crouzeix-Raviart and Morley elements. All methods are shown to be quasi-optimal, with quasi-optimality constant bounded in terms of shape regularity, and computationally feasible. In the third part of the thesis we restrict our attention to the two-dimensional Poisson problem and compare the performance of different quasi-optimal, first-order methods on various benchmarks. The purpose of these tests is twofold. On the one hand, we aim at assessing the actual size of the constants involved in our analysis. On the other hand, we highlight the importance of the proposed notions of stability and consistency when rough load terms come into play. All the numerical experiments are implemented within the finite element toolbox ALBERTA.