Div First-Order System LL* (FOSLL*) for Second-Order Elliptic Partial Differential Equations

The first-order system LL* (FOSLL*) approach for general second-order elliptic partial differential equations was proposed and analyzed in [10], in order to retain the full efficiency of the L2 norm first-order system least-squares (FOSLS) ap- proach while exhibiting the generality of the inverse-norm FOSLS approach. The FOSLL* approach in [10] was applied to the div-curl system with added slack vari- ables, and hence it is quite complicated. In this paper, we apply the FOSLL* approach to the div system and establish its well-posedness. For the corresponding finite ele- ment approximation, we obtain a quasi-optimal a priori error bound under the same regularity assumption as the standard Galerkin method, but without the restriction to sufficiently small mesh size. Unlike the FOSLS approach, the FOSLL* approach does not have a free a posteriori error estimator, we then propose an explicit residual error estimator and establish its reliability and efficiency bound

The DPG-star method

This article introduces the DPG-star (from now on, denoted DPG$^*$) finite element method. It is a method that is in some sense dual to the discontinuous Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG$^*$ methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddle-point problems. The analyses of the two problems have many common elements. Comparison to other methods in the literature round out the newly garnered perspective. Notably, DPG$^*$ and DPG methods can be seen as generalizations of $\mathcal{L}\mathcal{L}^\ast$ and least-squares methods, respectively. A priori error analysis and a posteriori error control for the DPG$^*$ method are considered in detail. Reports of several numerical experiments are provided which demonstrate the essential features of the new method. A notable difference between the results from the DPG$^*$ and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable

Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems

The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov $n$-widths of the solution sets. The central ingredient is the construction of computationally feasible "tight" surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters

A DPG method for linear quadratic optimal control problems

The DPG method with optimal test functions for solving linear quadratic optimal control problems with control constraints is studied. We prove existence of a unique optimal solution of the nonlinear discrete problem and characterize it through first order optimality conditions. Furthermore, we systematically develop a priori as well as a posteriori error estimates. Our proposed method can be applied to a wide range of constrained optimal control problems subject to, e.g., scalar second-order PDEs and the Stokes equations. Numerical experiments that illustrate our theoretical findings are presented

Least-Squares FEM: Literature Review

During the last years the interest in least squares finite element methods (LSFEM) has grown continuously. Least squares finite element methods offer some advantages over the widely used Galerkin variational principle. One reason is the ability to cope with first order differential operators without special treatment as required by the Galerkin FEM. The other reason comes from the numerical point of view, where the LSFEM leads to symmetric positive definite matrices which can be solved very efficiently under some conditions. This report gives an overview about the recent literature which appeared in the field of least squares finite element methods and summarises the essential results and facts about the LSFEM.Während der letzten Jahre hat das Interesse an Least Squares Finite Element Methoden (LSFEM) stetig zugenommen. Least Squares Finite Element Methoden bieten einige Vorteile gegenüber dem etablierten Galerkin Variationsansatz. So können Differentialoperatoren erster Ordnung ohne besondere numerische Techniken, wie z.B. Stabilisierung, direkt behandelt werden. Ein anderer Grund für den Einsatz der LSFEM liegt in den entstehenden algebraischen Gleichungssystemen, die immer symmetrisch positiv definit sind und unter bestimmten Vorraussetzungen eine effiziente Lösung ermöglichen.Dieser Bericht gibt einen Überblick über die aktuelle Literatur zur LSFEM und faßt die entscheidenden Ergebnisse zusammen

A pollution-free ultra-weak FOSLS discretization of the Helmholtz equation

We consider an ultra-weak first order system discretization of the Helmholtz equation. By employing the optimal test norm, the `ideal' method yields the best approximation to the pair of the Helmholtz solution and its scaled gradient w.r.t.~the norm on $L_2(\Omega)\times L_2(\Omega)^d$ from the selected finite element trial space. On convex polygons, the `practical', implementable method is shown to be pollution-free when the polynomial degree of the finite element test space grows proportionally with $\log \kappa$. Numerical results also on other domains show a much better accuracy than for the Galerkin method