Least-Squares Finite Element Methods: Nonconforming Methods and Augmented Mixed Methods

Abstract

Least-squares finite element methods (LSFEMs) represent an effort to preserve the benefits of the Rayleigh–Ritz framework for a broader range of problems. Essentially, they provide a way to maintain the advantages of this approach for nearly any first-order partial differential equation (PDE) problem, but also provide a useful tool, the a posterior estimator, for the adaptive finite element methods, as long as the norm equivalence of the least-squares functional is verified.

In the first part of this thesis, LSFEMs for general second-order elliptic equations with nonconforming finite element approximations. The equation may be indefinite. The degrees of freedom in nonconforming finite elements differ from those in conforming finite elements, allowing the bypassing of normal definition over the nodal points of approximated boundaries in curved boundary problems such as the slip-boundary conditions for the Stokes equation. However, nonconforming LSFEMs are still not well studied. Even though the diffusion problem without the convection and the reaction terms is studied in Duanhuo Yuan's 2003 Math. Comp. paper, the general elliptic equation and the least-squares estimate are not available because the norm equivalence for the error function does not work for the standard LS functional, where a counter example is given in this thesis. To this end, for the two-field potential-flux div LSFEMs with Crouzeix-Raviart (CR) element approximation, we present three proofs of the discrete solvability under the condition that mesh size is small enough. One of the proof is based on the coerciveness of the original bilinear form. The other two are based on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. A counterexample shows that div least-squares functional does not have norm equivalence in the sum space of H¹ and CR finite element spaces. Thus it cannot be used as an a posteriori error estimator. Several versions of reliable and efficient error estimators are proposed for the method. We also propose a three-filed potential-flux-intensity div-curl least-squares method with general nonconforming finite element approximations. The norm equivalence in the abstract nonconforming piecewise H¹-space is established for the three-filed formulation on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. The three-filed div-curl nonconforming formulation thus has no restriction on the mesh size, and the least-squares functional can be used as the built-in a posteriori error estimator. Under some restrictive conditions, we also discuss a potential-flux div-curl least-squares method.

In the second part of this thesis, for the generalized Darcy problem (an elliptic equation with discontinuous coefficients), we study a special Galerkin-Least-Squares method, known as the augmented mixed finite element method, and its relationship to the standard LSFEMs. For the classic LSFEMs, the energy estimate relies on the Poincaré-type inequality (Korn's inequality in the Stokes problem and the elasticity problem) and hence in the sense of robustness (the dependence of the diffusion coefficient) of the error estimators of the classic LSFEMs is not available. The new Galerkin-LSFEMs is an extension of the classic LSFEMs to solve the problem of the robustness in the a posterior estimate, meaning that the efficiency and the reliability of the error estimator should be not affected by some key parameters, the diffusion coefficient or perhaps the polynomial degrees and so on. To this end, two versions of augmented mixed finite element methods are proposed in the paper with robust a priori and a posteriori error estimates. As partial least-squares methods, the augmented mixed finite element methods are more flexible than the original least-squares finite element methods. The a posteriori error estimators of the augmented mixed finite element methods are the evaluations of the numerical solutions at the corresponding least-squares functionals. As comparisons, we discuss the non-robustness of the closely related least-squares finite element methods. Numerical experiments are presented to verify our findings.

In the final part of the thesis, we discuss some applications and the future studies for the theory and the proposed LSFEMs of this thesis, including the CR LSFEMs for the Stokes equation with the slip-boundary condition and the augmented mixed method for the Stokes interface problem.

Date of Award	1 Aug 2024
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Shun ZHANG (Supervisor)