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

Quasi-optimal nonconforming methods for symmetric elliptic problems. I -- Abstract theory

We consider nonconforming methods for symmetric elliptic problems and characterize their quasi-optimality in terms of suitable notions of stability and consistency. The quasi-optimality constant is determined and the possible impact of nonconformity on its size is quantified by means of two alternative consistency measures. Identifying the structure of quasi-optimal methods, we show that their construction reduces to the choice of suitable linear operators mapping discrete functions to conforming ones. Such smoothing operators are devised in the forthcoming parts of this work for various finite element spaces

Quasi-optimal nonconforming methods for symmetric elliptic problems. III-discontinuous Galerkin and other interior penalty methods

We devise new variants of the following nonconforming finite element methods: discontinuous Galerkin methods of fixed arbitrary order for the Poisson problem, the Crouzeix-Raviart interior penalty method for linear elasticity, and the quadratic C0 interior penalty method for the biharmonic problem. Each variant differs from the original method only in the discretization of the right-hand side. Before applying the load functional, a linear operator transforms nonconforming discrete test functions into conforming functions such that stability and consistency are improved. The new variants are thus quasi-optimal with respect to an extension of the energy norm. Furthermore, their quasi-optimality constants are uniformly bounded for shape regular meshes and tend to 1 as the penalty parameter increases

Quasi-optimal nonconforming methods for symmetric elliptic problems. I—abstract theory

We consider nonconforming methods for symmetric elliptic problems and characterize their quasi-optimality in terms of suitable notions of stability and consistency. The quasi-optimality constant is determined, and the possible impact of nonconformity on its size is quantified by means of two alternative consistency measures. Identifying the structure of quasi-optimal methods, we show that their construction reduces to the choice of suitable linear operators mapping discrete functions to conforming ones. Such smoothing operators are devised in the forthcoming parts of this work for various finite element spaces

Quasi-optimal Discontinuous Galerkin discretisations of the pp-Dirichlet problem

The classical arguments employed when obtaining error estimates of Finite Element (FE) discretisations of elliptic problems lead to more restrictive assumptions on the regularity of the exact solution when applied to non-conforming methods. The so-called minimal regularity estimates available in the literature relax some of these assumptions, but are not truly of -minimal regularity-, since a data oscillation term appears in the error estimate. Employing an approach based on a smoothing operator, we derive for the first time error estimates for Discontinuous Galerkin (DG) type discretisations of non-linear problems with $(p,\delta)$-structure that only assume the natural $W^{1,p}$-regularity of the exact solution, and which do not contain any oscillation terms

Interpolation Operator on negative Sobolev Spaces

We introduce a Scott--Zhang type projection operator mapping to Lagrange elements for arbitrary polynomial order. In addition to the usual properties, this operator is compatible with duals of first order Sobolev spaces. More specifically, it is stable in the corresponding negative norms and allows for optimal rates of convergence. We discuss alternative operators with similar properties. As applications of the operator we prove interpolation error estimates for parabolic problems and smoothen rough right-hand sides in a least squares finite element method