A weak Galerkin finite element method for second-order elliptic problems
01 Mar 2013-Journal of Computational and Applied Mathematics (North-Holland)-Vol. 241, Iss: 1, pp 103-115
TL;DR: A finite element method by using a weakly defined gradient operator over generalized functions and an optimal order error estimate in both a discrete H^1 and L^2 norms are established for the corresponding weak Galerkin finite element solutions.
About: This article is published in Journal of Computational and Applied Mathematics.The article was published on 2013-03-01 and is currently open access. It has received 497 citations till now. The article focuses on the topics: Discontinuous Galerkin method & Galerkin method.
Citations
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TL;DR: In this article, a weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations.
Abstract: . 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.
440 citations
TL;DR: In this article, an arbitrary-order locking-free method for linear elasticity is proposed, which relies on a pure-displacement (primal) formulation and leads to a symmetric, positive definite system matrix with compact stencil.
320 citations
TL;DR: In this article, the authors considered the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach, and they used the L2-projection operators as designed in [B. Ahmad, A. Beirao da Veiga, F. Brezzi and A. Russo.
Abstract: 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.
305 citations
Additional excerpts
...See [22], [30], [31], [32], , [33], [35], [36], [37], [39], [55], [54], [56], [68], [69]....
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TL;DR: In this article, a weak Galerkin (WG) finite element method for the Stokes equations in the primal velocity-pressure formulation is introduced. But this method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.
Abstract: This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primal velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k?1 for the velocity and polynomials of degree k?1 for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree k?1 on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.
234 citations
TL;DR: An evolution of the virtual elements of minimal degree for the approximation of the Cahn--Hilliard equation is developed and the convergence of the semidiscrete scheme is proved and the performance of the fully discrete scheme is investigated through a set of numerical tests.
Abstract: In this paper we develop an evolution of the $C^1$ virtual elements of minimal degree for the approximation of the Cahn--Hilliard equation. The proposed method has the advantage of being conforming in $H^2$ and making use of a very simple set of degrees of freedom, namely, 3 degrees of freedom per vertex of the mesh. Moreover, although the present method is new also on triangles, it can make use of general polygonal meshes. As a theoretical and practical support, we prove the convergence of the semidiscrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests.
219 citations
Cites background from "A weak Galerkin finite element meth..."
...Other virtual element contributions are, for instance, [11, 3, 5, 8, 14, 27, 37, 38], while for a very short sample of other FEM-inspired methods dealing with general polygons we refer to [10, 16, 18, 19, 26, 28, 41, 42, 46, 47]....
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References
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Book•
01 Jan 1978TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Abstract: Preface 1. Elliptic boundary value problems 2. Introduction to the finite element method 3. Conforming finite element methods for second-order problems 4. Other finite element methods for second-order problems 5. Application of the finite element method to some nonlinear problems 6. Finite element methods for the plate problem 7. A mixed finite element method 8. Finite element methods for shells Epilogue Bibliography Glossary of symbols Index.
8,407 citations
Book•
01 Apr 2002
TL;DR: In this article, Ciarlet presents a self-contained book on finite element methods for analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces.
Abstract: From the Publisher:
This book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces. Other than these basics, the book is mathematically self-contained.
About the Author
Philippe G. Ciarlet is a Professor at the Laboratoire d'Analyse Numerique at the Universite Pierre et Marie Curie in Paris. He is also a member of the French Academy of Sciences. He is the author of more than a dozen books on a variety of topics and is a frequent invited lecturer at meetings and universities throughout the world. Professor Ciarlet has served approximately 75 visiting professorships since 1973, and he is a member of the editorial boards of more than 20 journals.
8,052 citations
"A weak Galerkin finite element meth..." refers methods in this paper
...nd ψ = ψ(x) – either vector-valued or scalar-valued functions. Here ∇udenotes the gradient of the function u= u(x), and ∇ is known as the gradient operator. In the standard Galerkin method (e.g., see [13, 7]), the trial space H1(Ω) and the test space H1 0(Ω) in (1.4) are each replaced by properly defined subspaces of finite dimensions. The resulting solution in the subspace/subset is called a Galerkin appr...
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.... Let Th be a triangular partition of the domain Ω with mesh size h. Assume that the partition Th is shape regular so that the routine inverse inequality in the finite element analysis holds true (see [13]). In the general spirit of Galerkin procedure, we shall design a weak Galerkin method for (4.1) by following two basic principles: (1) replace H1(Ω) by a space of discrete weak functions defined on th...
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...classified into two categories: (1) methods based on the primary variable u, and (2) methods based on the variable uand a flux variable (mixed formulation). The standard Galerkin finite element methods ([13, 7, 5]) and various interior penalty type discontinuous Galerkin methods ([1, 3, 6, 21, 22]) are 3 typical examples of the first category. The standard mixed finite elements ([20, 2, 4, 8, 9, 11, 10, 24]) and...
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Book•
14 Feb 2013
TL;DR: In this article, the construction of a finite element of space in Sobolev spaces has been studied in the context of operator-interpolation theory in n-dimensional variational problems.
Abstract: Preface(2nd ed.).- Preface(1st ed.).- Basic Concepts.- Sobolev Spaces.- Variational Formulation of Elliptic Boundary Value Problems.- The Construction of a Finite Element of Space.- Polynomial Approximation Theory in Sobolev Spaces.- n-Dimensional Variational Problems.- Finite Element Multigrid Methods.- Additive Schwarz Preconditioners.- Max-norm Estimates.- Adaptive Meshes.- Variational Crimes.- Applications to Planar Elasticity.- Mixed Methods.- Iterative Techniques for Mixed Methods.- Applications of Operator-Interpolation Theory.- References.- Index.
7,158 citations
"A weak Galerkin finite element meth..." refers methods in this paper
...This research was supported in part by National Science Foundation Grant DMS0813571 1 2 where (φ, ψ) represents the L2-inner product of φ = φ(x) and ψ = ψ(x) – either vector-valued or scalar-valued functions....
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...The existing methods can be classified into two categories: (1) methods based on the primary variable u, and (2) methods based on the variable u and a flux variable (mixed formulation)....
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Book•
23 Nov 2011TL;DR: Variational Formulations and Finite Element Methods for Elliptic Problems, Incompressible Materials and Flow Problems, and Other Applications.
Abstract: Variational Formulations and Finite Element Methods. Approximation of Saddle Point Problems. Function Spaces and Finite Element Approximations. Various Examples. Complements on Mixed Methods for Elliptic Problems. Incompressible Materials and Flow Problems. Other Applications.
5,030 citations
TL;DR: In this paper, a framework for the analysis of a large class of discontinuous Galerkin methods for second-order elliptic problems is provided, which allows for the understanding and comparison of most of the discontinuous methods that have been proposed over the past three decades.
Abstract: We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.
3,293 citations