Hybrid First-Order System Least-Squares Finite Element Methods With The Application To Stokes And Navier-Stokes Equations

This thesis combines the FOSLS method with the FOSLL* method to create a Hybrid method. The FOSLS approach minimizes the error, e h = uh − u, over a finite element subspace, [special characters omitted], in the operator norm, [special characters omitted] ||L(uh − u)||. The FOSLL* method looks for an approximation in the range of L*, setting uh = L*wh and choosing wh ∈ [special characters omitted], a standard finite element space. FOSLL* minimizes the L 2 norm of the error over L*([special characters omitted]), that is, [special characters omitted] ||L*wh − u||. FOSLS enjoys a locally sharp, globally reliable, and easily computable a posterior error estimate, while FOSLL* does not. The Hybrid method attempts to retain the best properties of both FOSLS and FOSLL*. This is accomplished by combining the FOSLS functional, the FOSLL* functional, and an intermediate term that draws them together. The Hybrid method produces an approximation, uh, that is nearly the optimal over [special characters omitted] in the graph norm, ||eh[special characters omitted] := ½||eh|| 2 + ||Leh|| 2. The FOSLS and intermediate terms in the Hybrid functional provide a very effective a posteriori error measure. In this dissertation we show that the Hybrid functional is coercive and continuous in graph-like norm with modest coercivity and continuity constants, c0 = 1/3 and c1 = 3; that both ||eh|| and ||L eh|| converge with rates based on standard interpolation bounds; and that, if LL* has full H2-regularity, the L2 error, ||eh||, converges with a full power of the discretization parameter, h, faster than the functional norm. Letting ũh denote the optimum over [special characters omitted] in the graph norm, we also show that if superposition is used, then ||uh − ũ h[special characters omitted] converges two powers of h faster than the functional norm. Numerical tests on are provided to confirm the efficiency of the Hybrid method and effectiveness of the a posteriori error measure

Mixed formulations for the convection-diffusion equation

This thesis explores the numerical stability of the stationary Convection-Diffusion-Reaction (CDR) equation in mixed form, where the second-order equation is expressed as two first-order equations using a second variable relating to a derivative of the primary variable. This first-order system uses either a total or diffusive flux formulation. Westart by numerically testing the unstabilised Douglas and Roberts classical discretisation of the mixed CDR equation using Raviart-Thomas elements. The results indicate that,as expected, for both total and diffusive flux, the stability of the formulation degrades dramatically as diffusion decreases.Next, we investigate stabilised formulations that are designed to improve the ability of the discrete problem to cope with problems containing layers. We test the Masud and Kwack method that uses Lagrangian elements but whose analysis has not been developed.We then significantly modify the formulation to allow us to prove existence of a solution and facilitate the analysis. Our new method, which uses total flux, is then tested for convergence with standard tests and found to converge satisfactorily over a range of values of diffusion.Another family of first-order methods called First-Order System of Least-Squares (FOSLS/LSFEM) is also investigated in relation to solving the CDR equation. These symmetric,elliptic methods do not require stabilisation but also do not cope well with sharp layers and small diffusion. Modifications have been proposed and this study includes aversion of Chen et al. which uses diffusive flux, imposing boundary conditions weakly in a weighted formulation.We test our new method against all the aforementioned methods, but we find that other methods do not cope well with layers in standard tests. Our method compares favourably with the standard Streamline-Upwind-Petrov-Galerkin method (SUPG/SDFEM), but overall is not a significant improvement. With further fine-tuning, our method could improve but it has more computational overhead than SUPG.This thesis explores the numerical stability of the stationary Convection-Diffusion-Reaction (CDR) equation in mixed form, where the second-order equation is expressed as two first-order equations using a second variable relating to a derivative of the primary variable. This first-order system uses either a total or diffusive flux formulation. Westart by numerically testing the unstabilised Douglas and Roberts classical discretisation of the mixed CDR equation using Raviart-Thomas elements. The results indicate that,as expected, for both total and diffusive flux, the stability of the formulation degrades dramatically as diffusion decreases.Next, we investigate stabilised formulations that are designed to improve the ability of the discrete problem to cope with problems containing layers. We test the Masud and Kwack method that uses Lagrangian elements but whose analysis has not been developed.We then significantly modify the formulation to allow us to prove existence of a solution and facilitate the analysis. Our new method, which uses total flux, is then tested for convergence with standard tests and found to converge satisfactorily over a range of values of diffusion.Another family of first-order methods called First-Order System of Least-Squares (FOSLS/LSFEM) is also investigated in relation to solving the CDR equation. These symmetric,elliptic methods do not require stabilisation but also do not cope well with sharp layers and small diffusion. Modifications have been proposed and this study includes aversion of Chen et al. which uses diffusive flux, imposing boundary conditions weakly in a weighted formulation.We test our new method against all the aforementioned methods, but we find that other methods do not cope well with layers in standard tests. Our method compares favourably with the standard Streamline-Upwind-Petrov-Galerkin method (SUPG/SDFEM), but overall is not a significant improvement. With further fine-tuning, our method could improve but it has more computational overhead than SUPG

Parallel Efficiency-based Adaptive Local Refinement

New adaptive local refinement (ALR) strategies are developed, the goal of which is to reach a given error tolerance with the least amount of computational cost. This strategy is especially attractive in the setting of a first-order system least-squares (FOSLS) finite element formulation in conjunction with algebraic multigrid (AMG) methods in the context of nested iteration (NI). To accomplish this, the refinement decisions are determined based on minimizing the predicted `accuracy-per-computational-cost\u27 efficiency (ACE). The nested iteration approach produces a sequence of refinement levels in which the error is equally distributed across elements on a relatively coarse grid. Once the solution is numerically resolved, refinement becomes nearly uniform. Efficiency of the algorithms are demonstrated through a 2D Poisson problem with steep gradients, and 2D reduced model of the incompressible, resistive magnetohydrodynamic (MHD) equations. Accommodations of the ALR strategies to parallel computer architectures involve a geometric binning strategy to reduce communication cost. Load balancing begins at very coarse levels. Elements and nodes are redistributed using parallel quad-tree structures and a space filling curve. This automatically ameliorates load balancing issues at finer levels. Numerical results produced on Frost, the NCAR/CU Blue Gene/L supercomputer, are presented for a 2D Poisson problem with steep gradients, a 2D backward facing step incompressible Stokes equations and Navier-Stokes equations. The NI-FOSL-AMG-ACE approach is able to provide highly resolved approximations to rapidly varying solutions using a small number of work units. Excellent weak and strong scalability of parallel ALR are demonstrated on up to 4,096 processors for problems with up to 15 million biquadratic elements

Efficiency-based hp-refinement for finite element methods

Two efficiency-based grid refinement strategies are investigated for adaptive finite element solution of partial differential equations. In each refinement step, the elements are ordered in terms of decreasing local error, and the optimal fraction of elements to be refined is deter- mined based on e±ciency measures that take both error reduction and work into account. The goal is to reach a pre-specified bound on the global error with a minimal amount of work. Two efficiency measures are discussed, 'work times error' and 'accuracy per computational cost'. The resulting refinement strategies are first compared for a one-dimensional model problem that may have a singularity. Modified versions of the efficiency strategies are proposed for the singular case, and the resulting adaptive methods are compared with a threshold-based refinement strategy. Next, the efficiency strategies are applied to the case of hp-refinement for the one-dimensional model problem. The use of the efficiency-based refinement strategies is then explored for problems with spatial dimension greater than one. The work times error strategy is inefficient when the spatial dimension, d, is larger than the finite element order, p, but the accuracy per computational cost strategy provides an efficient refinement mechanism for any combination of d and p

Dual-Norm Least-Squares Finite Element Methods for Hyperbolic Problems

Least-squares finite element discretizations of first-order hyperbolic partial differential equations (PDEs) are proposed and studied. Hyperbolic problems are notorious for possessing solutions with jump discontinuities, like contact discontinuities and shocks, and steep exponential layers. Furthermore, nonlinear equations can have rarefaction waves as solutions. All these contribute to the challenges in the numerical treatment of hyperbolic PDEs. The approach here is to obtain appropriate least-squares formulations based on suitable minimization principles. Typically, such formulations can be reduced to one or more (e.g., by employing a Newton-type linearization procedure) quadratic minimization problems. Both theory and numerical results are presented. A method for nonlinear hyperbolic balance and conservation laws is proposed. The formulation is based on a Helmholtz decomposition and closely related to the notion of a weak solution and a $H^{-1}$-type least-squares principle. Accordingly, the respective important conservation properties are studied in detail and the theoretically challenging convergence properties, with respect to the $L^2$ norm, are discussed. In the linear case, the convergence in the $L^2$ norm is explicitly and naturally guaranteed by suitable formulations that are founded upon the original \cL\cL^\star method developed for elliptic PDEs. The approaches considered here are the \cL\cL^\star-based and (\cL\cL^\star)^{-1} methods, where the latter utilizes a special negative-norm least-squares minimization principle. These methods can be viewed as specific approximations of the generally infeasible quadratic minimization that determines the $L^2$-orthogonal projection of the exact solution. The formulations are analyzed and studied in detail. Key words: first-order hyperbolic problems; hyperbolic balance laws; Burgers equation; weak solutions; Helmholtz decomposition; conservation properties; space-time discretization; least-squares methods; dual methods; adjoint methods; negative-norm methods; finite element methods; discontinuous coefficients; exponential layers AMS subject classifications: 65N30, 65N1

Dual-Norm Least-Squares Finite Element Methods for Hyperbolic Problems

Least-squares finite element discretizations of first-order hyperbolic partial differential equations (PDEs) are proposed and studied.Hyperbolic problems are notorious for possessing solutions with jump discontinuities, like contact discontinuities and shocks, and steep exponential layers. Furthermore, nonlinear equations can have rarefaction waves as solutions. All these contribute to the challenges in the numerical treatment of hyperbolic PDEs.&nbsp; The approach here is to obtain appropriate least-squares formulations based on suitable minimization principles. Typically, such formulations can be reduced to one or more (e.g., by employing a Newton-type linearization procedure) quadratic minimization problems. Both theory and numerical results are presented. A method for nonlinear hyperbolic balance and conservation laws is proposed. The formulation is based on a Helmholtz decomposition and closely related to the notion of a weak solution and a H⁻&sup1;-type least-squares principle. Accordingly, the respective important conservation properties are studied in detail and the theoretically challenging convergence properties, with respect to the L&sup2; norm, are discussed. In the linear case, the convergence in the L&sup2; norm is explicitly and naturally guaranteed by suitable formulations that are founded upon the original LL* method developed for elliptic PDEs. The approaches considered here are the LL*-based and LL*⁻&sup1; methods, where the latter utilizes a special negative-norm least-squares minimization principle. These methods can be viewed as specific approximations of the generally infeasible quadratic minimization that determines the L&sup2;-orthogonal projection of the exact solution. The formulations are analyzed and studied in detail. &nbsp;</p

A survey of Trefftz methods for the Helmholtz equation

Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces. We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions; we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties. One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.Comment: 41 pages, 2 figures, to appear as a chapter in Springer Lecture Notes in Computational Science and Engineering. Differences from v1: added a few sentences in Sections 2.1, 2.2.2 and 2.3.1; inserted small correction

Least-squares methods for computational electromagnetics

The modeling of electromagnetic phenomena described by the Maxwell's equations is of critical importance in many practical applications. The numerical simulation of these equations is challenging and much more involved than initially believed. Consequently, many discretization techniques, most of them quite complicated, have been proposed. In this dissertation, we present and analyze a new methodology for approximation of the time-harmonic Maxwell's equations. It is an extension of the negative-norm least-squares finite element approach which has been applied successfully to a variety of other problems. The main advantages of our method are that it uses simple, piecewise polynomial, finite element spaces, while giving quasi-optimal approximation, even for solutions with low regularity (such as the ones found in practical applications). The numerical solution can be efficiently computed using standard and well-known tools, such as iterative methods and eigensolvers for symmetric and positive definite systems (e.g. PCG and LOBPCG) and reconditioners for second-order problems (e.g. Multigrid). Additionally, approximation of varying polynomial degrees is allowed and spurious eigenmodes are provably avoided. We consider the following problems related to the Maxwell's equations in the frequency domain: the magnetostatic problem, the electrostatic problem, the eigenvalue problem and the full time-harmonic system. For each of these problems, we present a natural (very) weak variational formulation assuming minimal regularity of the solution. In each case, we prove error estimates for the approximation with two different discrete least-squares methods. We also show how to deal with problems posed on domains that are multiply connected or have multiple boundary components. Besides the theoretical analysis of the methods, the dissertation provides various numerical results in two and three dimensions that illustrate and support the theory