The MINI mixed finite element for the Stokes problem: An experimental investigation

Super-convergence of order 1.5 in pressure and velocity has been experimentally investigated for the two-dimensional Stokes problem discretised with the MINI mixed finite element. Even though the classic mixed finite element theory for the MINI element guarantees linear convergence for the total error, recent theoretical results indicate that super-convergence of order 1.5 in pressure and of the linear part of the computed velocity to the piecewise linear nodal interpolation of the exact velocity is in fact possible with structured, three-directional triangular meshes. The numerical experiments presented here suggest a more general validity of super-convergence of order 1.5, possibly to automatically generated and unstructured triangulations. In addition, the approximating properties of the complete computed velocity have been compared with the approximating properties of the piecewise-linear part of the computed velocity, finding that the former is generally closer to the exact velocity, whereas the latter conserves mass better

Comparison results for the Stokes equations

This paper enfolds a medius analysis for the Stokes equations and compares different finite element methods (FEMs). A first result is a best approximation result for a P1 non-conforming FEM. The main comparison result is that the error of the P2-P0-FEM is a lower bound to the error of the Bernardi-Raugel (or reduced P2-P0) FEM, which is a lower bound to the error of the P1 non-conforming FEM, and this is a lower bound to the error of the MINI-FEM. The paper discusses the converse direction, as well as other methods such as the discontinuous Galerkin and pseudostress FEMs. Furthermore this paper provides counterexamples for equivalent convergence when different pressure approximations are considered. The mathematical arguments are various conforming companions as well as the discrete inf-sup condition

Stable and locally mass- and momentum-conservative control-volume finite-element schemes for the Stokes problem

We introduce new control-volume finite-element discretization schemes suitable for solving the Stokes problem. Within a common framework, we present different approaches for constructing such schemes. The first and most established strategy employs a non-overlapping partitioning into control volumes. The second represents a new idea by splitting into two sets of control volumes, the first set yielding a partition of the domain and the second containing the remaining overlapping control volumes required for stability. The third represents a hybrid approach where finite volumes are combined with finite elements based on a hierarchical splitting of the ansatz space. All approaches are based on typical finite element function spaces but yield locally mass and momentum conservative discretization schemes that can be interpreted as finite volume schemes. We apply all strategies to the inf-sub stable MINI finite-element pair. Various test cases, including convergence tests and the numerical observation of the boundedness of the number of preconditioned Krylov solver iterations, as well as more complex scenarios of flow around obstacles or through a three-dimensional vessel bifurcation, demonstrate the stability and robustness of the schemes

A tangential and penalty-free finite element method for the surface Stokes problem

Surface Stokes and Navier-Stokes equations are used to model fluid flow on surfaces. They have attracted significant recent attention in the numerical analysis literature because approximation of their solutions poses significant challenges not encountered in the Euclidean context. One challenge comes from the need to simultaneously enforce tangentiality and $H^1$ conformity (continuity) of discrete vector fields used to approximate solutions in the velocity-pressure formulation. Existing methods in the literature all enforce one of these two constraints weakly either by penalization or by use of Lagrange multipliers. Missing so far is a robust and systematic construction of surface Stokes finite element spaces which employ nodal degrees of freedom, including MINI, Taylor-Hood, Scott-Vogelius, and other composite elements which can lead to divergence-conforming or pressure-robust discretizations. In this paper we construct surface MINI spaces whose velocity fields are tangential. They are not $H^1$-conforming, but do lie in $H({\rm div})$ and do not require penalization to achieve optimal convergence rates. We prove stability and optimal-order energy-norm convergence of the method and demonstrate optimal-order convergence of the velocity field in $L_2$ via numerical experiments. The core advance in the paper is the construction of nodal degrees of freedom for the velocity field. This technique also may be used to construct surface counterparts to many other standard Euclidean Stokes spaces, and we accordingly present numerical experiments indicating optimal-order convergence of nonconforming tangential surface Taylor-Hood $\mathbb{P}^2-\mathbb{P}^1$ elements

Analytical investigations and numerical experiments for singularly perturbed convection-diffusion problems with layers and singularities using a newly developed FE-software

In the field of singularly perturbed reaction- or convection-diffusion boundary value problems the research area of a priori error analysis for the finite element method, has already been thoroughly investigated. In particular, for mesh adapted methods and/or various stabilization techniques, works have been done that prove optimal rates of convergence or supercloseness uniformly in the perturbation parameter epsilon. Commonly, however, it is assumed that the exact solution behaves nicely in that it obeys certain regularity assumptions although in general, e.g. due to corner singularities, these regularity requirements are not satisfied. So far, insufficient regularity has been met by assuming compatibility conditions on the data. The present thesis originated from the question: What can be shown if these rather unrealistic additional assumptions are dropped? We are interested in epsilon-uniform a priori estimates for convergence and superconvergence that include some regularity parameter that is adjustable to the smoothness of the exact solution. A major difficulty that occurs when seeking the numerical error decay is that the exact solution is not known. Since we strive for reliable rates of convergence we want to avoid the standard approach of the "double-mesh principle". Our choice is to use reference solutions as a substitute for the exact solution. Numerical experiments are intended to confirm the theoretical results and to bring further insights into the interplay between layers and singularities. To computationally realize the thereby arising demanding practical aspects of the finite element method, a new software is developed that turns out to be particularly suited for the needs of the numerical analyst. Its design, features and implementation is described in detail in the second part of the thesis

A new framework for the analysis of finite element methods for fluid-structure interaction problems

Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure's displacement have been extensively studied for incompressible fluid-structure interaction (FSI) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions to the FSI problems in the standard $L^2$ norm. In this article, we propose a new kinematically coupled scheme for incompressible FSI thin-structure model and establish a new framework for the numerical analysis of FSI problems in terms of a newly introduced coupled non-stationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the $L^2$ norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field

A Multigrid Method for the Efficient Numerical Solution of Optimization Problems Constrained by Partial Differential Equations

We study the minimization of a quadratic functional subject to constraints given by a linear or semilinear elliptic partial differential equation with distributed control. Further, pointwise inequality constraints on the control are accounted for. In the linear-quadratic case, the discretized optimality conditions yield a large, sparse, and indefinite system with saddle point structure. One main contribution of this thesis consists in devising a coupled multigrid solver which avoids full constraint elimination. To this end, we define a smoothing iteration incorporating elements from constraint preconditioning. A local mode analysis shows that for discrete optimality systems, we can expect smoothing rates close to those obtained with respect to the underlying constraint PDE. Our numerical experiments include problems with constraints where standard pointwise smoothing is known to fail for the underlying PDE. In particular, we consider anisotropic diffusion and convection-diffusion problems. The framework of our method allows to include line smoothers or ILU-factorizations, which are suitable for such problems. In all cases, numerical experiments show that convergence rates do not depend on the mesh size of the finest level and discrete optimality systems can be solved with a small multiple of the computational cost which is required to solve the underlying constraint PDE. Employing the full multigrid approach, the computational cost is proportional to the number of unknowns on the finest grid level. We discuss the role of the regularization parameter in the cost functional and show that the convergence rates are robust with respect to both the fine grid mesh size and the regularization parameter under a mild restriction on the next to coarsest mesh size. Incorporating spectral filtering for the reduced Hessian in the control smoothing step allows us to weaken the mesh size restriction. As a result, problems with near-vanishing regularization parameter can be treated efficiently with a negligible amount of additional computational work. For fine discretizations, robust convergence is obtained with rates which are independent of the regularization parameter, the coarsest mesh size, and the number of levels. In order to treat linear-quadratic problems with pointwise inequality constraints on the control, the multigrid approach is modified to solve subproblems generated by a primal-dual active set strategy (PDAS). Numerical experiments demonstrate the high efficiency of this approach due to mesh-independent convergence of both the outer PDAS method and the inner multigrid solver. The PDAS-multigrid method is incorporated in the sequential quadratic programming (SQP) framework. Inexact Newton techniques further enhance the computational efficiency. Globalization is implemented with a line search based on the augmented Lagrangian merit function. Numerical experiments highlight the efficiency of the resulting SQP-multigrid approach. In all cases, locally superlinear convergence of the SQP method is observed. In combination with the mesh-independent convergence rate of the inner solver, a solution method with optimal efficiency is obtained

Advanced Arbitrary Lagrangian-Eulerian Finite Element Methods for Unsteady Multiphysics Problems Involving Moving Interfaces/Boundaries

In this dissertation, two kinds of arbitrary Lagrangian-Eulerian (ALE)-finite element methods (FEM) within the monolithic approach are studied for unsteady multiphysics coupling problems involving the moving interfaces/boundaries. For the classical affine-type ALE mapping that is studied in the first part of this dissertation, we develop the monolithic ALE-FEM and conduct stability and optimal convergence analyses in the energy norm for the transient Stokes/parabolic interface problem with jump coefficients, and more realistically, for the dynamic fluid-structure interaction (FSI) problems by taking the discrete ALE mapping and the discrete mesh velocity into a careful consideration of our numerical analyses and computations, where the affine-type ALE mapping preserves $H^1$-invariance for both the Stokes (fluid) equations and the parabolic (structure) equation in their moving subdomains all the time. In particular, we analyze the ALE-FEM for Stokes/parabolic interface problem by introducing a novel mixed-type $H^1$-projection with a moving interface and the discrete mesh velocity. We first obtain the well-posedness and convergence properties for this new $H^1$-projection and its ALE time derivative, by means of which we then derive the optimal error estimate in the energy norm and the sup-optimal error estimate in $L^2$ norm for both semi- and fully discrete mixed finite element approximations to the Stokes/parabolic interface problem. As for the realistic FSI problem, we build the classical affine-type ALE mapping into our novel mixed-type $H^1$-projection that couples the Eulerian fluid equation and the Lagrangian structure equation through a moving interface, and study its well-posedness and optimal convergence properties. Then we are able to analyze the (nearly) optimal error estimate in various norms for the ALE-finite element approximation to FSI problem as well. In the second part of this dissertation, a novel Piola-type ALE mapping and the associated ALE-FEM are developed and are well analyzed for two types of moving interface problems whose weak forms are associated with $H(\text{div})$ space: the mixed parabolic problem in a moving domain, and the mixed parabolic/parabolic moving interface problem. In practice, the multiphysics problems involving the pore (Darcy\u27s) fluid equation, or more sophisticatedly, the poroelasticity (Biot\u27s) model, which may stay alone in a moving domain or interact with other field models through a moving interface, essentially belong to these two types of problems that we study in this part. The key idea of the developed Piola-type ALE mapping is to preserve $H(\text{div})$-invariance with time for the moving interfaces/boundaries problems that are associated with $H(\text{div})$ space in moving (sub)domains. Utilizing a specific stabilization technique, we apply the stable Stokes-pair to the mixed ALE-finite element discretization of both problems, design their semi- and fully discrete Piola-type ALE-finite element schemes, and analyze their stability and optimal convergence results using the MINI mixed element. All theoretical results obtained in this dissertation are appropriately validated by our numerical experiments using various numerical examples