Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

. A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. Allowing the use of discontinuous approximating functions on arbitrary shape of polygons/polyhedra makes the method highly flexible in practical computation. Optimal order error estimates in both discrete H and L norms are established for the corresponding weak Galerkin mixed finite element solutions.

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A weak Galerkin mixed finite element method for second order elliptic problems

The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This article reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, $H(div)$-conforming finite ...

On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows

Adaptive finite element methods (FEM) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

Keywords: A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, Stokes, Uzawa

AMS Subject Classifications: 65N12, 65N15, 65N30, 65N50, 65Y20


Published: SIAM Review, 44 (2002) 631--658.

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Convergence of Adaptive Finite Element Methods

We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [A. H. Schatz, An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comput. 28 (1974) 959–962] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214]), we use here, in a systematic way, the L2-projection operators as designed in [B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013) 376–391]. In particular, the present method does not reduce to the original Virtual Element Method of [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] for simpler problems as the classical Laplace operator (apart from the lowest-order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] to the case of variable coefficients produces, in general, sub-optimal results.

Virtual Element Methods for general second order elliptic problems on polygonal meshes

The weak Galerkin finite element method is a novel numerical method that was first proposed and analyzed by Wang and Ye (2011) for general second order elliptic problems on triangular meshes. The goal of this paper is to conduct a computational investigation for the weak Galerkin method for various model problems with more general finite element partitions. The numerical results confirm the theory established in Wang and Ye (2011). The results also indicate that the weak Galerkin method is efficient, robust, and reliable in scientific computing.

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A computational study of the weak Galerkin method for second-order elliptic equations

The weak Galerkin finite element method is a novel numerical method that was first proposed and analyzed by Wang and Ye for general second order elliptic problems on triangular meshes. The goal of this paper is to conduct a computational investigation for the weak Galerkin method for various model problems with more general finite element partitions. The numerical results confirm the theory established by Wang and Ye. The results also indicate that the weak Galerkin method is efficient, robust, and reliable in scientific computing.

A Computational Study of the Weak Galerkin Method for Second-Order Elliptic Equations

A family of weak Galerkin finite element discretization is developed for solving the coupled Darcy-Stokes equation. The equation in consideration admits the Beaver-Joseph-Saffman condition on the interface. By using the weak Galerkin approach, in the discrete space we are able to impose the normal continuity of velocity explicitly. Or in other words, strong coupling is achieved in the discrete space. Different choices of weak Galerkin finite element spaces are discussed, and error estimates are given.

Weak Galerkin method for the coupled Darcy–Stokes flow

This article introduces and analyzes a weak Galerkin mixed finite element method for solving the biharmonic equation The weak Galerkin method, first introduced by two of the authors (J Wang and X Ye) in (Wang et al, Comput Appl Math 241:103–115, 2013) for second-order elliptic problems, is based on the concept of discrete weak gradients The method uses completely discrete finite element functions, and, using certain discrete spaces and with stabilization, it works on partitions of arbitrary polygon or polyhedron In this article, the weak Galerkin method is applied to discretize the Ciarlet–Raviart mixed formulation for the biharmonic equation In particular, an a priori error estimation is given for the corresponding finite element approximations The error analysis essentially follows the framework of Babuska, Osborn, and Pitkaranta (Math Comp 35:1039–1062, 1980) and uses specially designed mesh-dependent norms The proof is technically tedious due to the discontinuous nature of the weak Galerkin finite element functions Some computational results are presented to demonstrate the efficiency of the method

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A Weak Galerkin Mixed Finite Element Method for Biharmonic Equations

A computational method based on a divergence-free $H$(div) approach is presented for the Stokes equations in this article. This method is designed to find velocity approximation in an exact divergence-free subspace of the corresponding $H$(div) finite element space. That is, the continuity equation is strongly enforced a priori and the pressure is eliminated from the linear system in calculation. A strength of this approach is that the saddle-point problem for Stokes equations is reduced to a symmetric positive definite problem in a subspace for which basis functions are readily available. The resulting discrete system can then be solved by using existing sophisticated solvers. The aim of this article is to demonstrate the efficiency and robustness of $H$(div) finite element methods for Stokes equations. The results not only confirm the existing theoretical results but also reveal additional advantages of the method in dealing with discontinuous boundary conditions.

Yanqiu Wang

Papers

A computational study of the weak Galerkin method for second-order elliptic equations

A Computational Study of the Weak Galerkin Method for Second-Order Elliptic Equations

Weak Galerkin method for the coupled Darcy–Stokes flow

A Weak Galerkin Mixed Finite Element Method for Biharmonic Equations

A Robust Numerical Method for Stokes Equations Based on Divergence-Free $H$(div) Finite Element Methods