Equivalent projectors for virtual element methods Ahmad, B.; Alsaedi, A.; Brezzi, F. ...
Computers & mathematics with applications (1987),
September 2013, 2013-09-00, 20130901, Letnik:
66, Številka:
3
Journal Article
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In the original virtual element space with degree of accuracy k, projector operators in the H1-seminorm onto polynomials of degree ≤k can be easily computed. On the other hand, projections in the L2 ...norm are available only on polynomials of degree ≤k−2 (directly from the degrees of freedom). Here, we present a variant of the virtual element method that allows the exact computations of the L2 projections on all polynomials of degree ≤k. The interest of this construction is illustrated with some simple examples, including the construction of three-dimensional virtual elements, the treatment of lower-order terms, the treatment of the right-hand side, and the L2 error estimates.
Basic principles of mixed Virtual Element Methods Brezzi, F.; Falk, Richard S.; Donatella Marini, L.
ESAIM. Mathematical modelling and numerical analysis,
07/2014, Letnik:
48, Številka:
4
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The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of H(div)-conforming vector fields (or, more generally, of (n ...− 1) − Cochains). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim of making the basic philosophy clear. However, we consider an arbitrary degree of accuracy k (the Virtual Element analogue of dealing with polynomials of arbitrary order in the Finite Element Framework).
We discuss the application of virtual elements to linear elasticity problems, for both the compressible and the nearly incompressible case. Virtual elements are very close to mimetic finite ...differences (see, for linear elasticity, L. Beirão da Veiga, M2AN Math. Model. Numer. Anal., 44 (2010), pp. 231-250) and in particular to higher order mimetic finite differences. As such, they share the good features of being able to represent in an exact way certain physical properties (conservation, incompressibility, etc.) and of being applicable in very general geometries. The advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity.
We give here a simplified presentation of the lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The ...method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic field H on each edge, and the vertex values of the Lagrange multiplier p (used to enforce the solenoidality of the magnetic induction B=μH). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called “first kind Nédélec” elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions.
We consider the use of nodal and edge Virtual Element spaces for the discretization of magnetostatic problems in two dimensions, following the variational formulation of Kikuchi. In addition, we ...present a novel Serendipity variant of the same spaces that allow to save many internal degrees of freedom. These Virtual Element Spaces of different type can be useful in applications where an exact sequence is particularly convenient, together with commuting-diagram interpolation operators, as is definitely the case in electromagnetic problems. We prove stability and optimal error estimates, and we check the performance with some academic numerical experiments.
•Virtual Element Methods generalize Finite Elements to polytopal geometries.•Serendipity VEMs imitate the Serendipity variants of Finite Elements.•On triangles we reproduce FE, on quads we have a ...more robust version of them.•On general polytopes, the Serendipity VEMs improve significantly the original ones.
We introduce a new variant of Nodal Virtual Element spaces that mimics the “Serendipity Finite Element Methods” (whose most popular example is the 8-node quadrilateral) and allows to reduce (often in a significant way) the number of internal degrees of freedom. When applied to the faces of a three-dimensional decomposition, this allows a reduction in the number of face degrees of freedom: an improvement that cannot be achieved by a simple static condensation. On triangular and tetrahedral decompositions the new elements (contrary to the original VEMs) reduce exactly to the classical Lagrange FEM. On quadrilaterals and hexahedra the new elements are quite similar (and have the same amount of degrees of freedom) to the Serendipity Finite Elements, but are much more robust with respect to element distortions. On more general polytopes the Serendipity VEMs are the natural (and simple) generalization of the simplicial case.
We extend the mimetic finite difference (MFD) method to the numerical treatment of magnetostatic fields problems in mixed
div–
curl form for the divergence-free magnetic vector potential. To ...accomplish this task, we introduce three sets of degrees of freedom that are attached to the vertices, the edges, and the faces of the mesh, and two discrete operators mimicking the curl and the gradient operator of the differential setting. Then, we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the
div–
curl variational formulation of the magnetostatic equations. This construction is based on an
algebraic consistency condition that generalizes the usual construction of the inner products of the MFD method. We also discuss the linear algebraic form of the resulting MFD scheme, its practical implementation, and discuss existence and uniqueness of the numerical solution by generalizing the concept of
logically rectangular or cubic meshes by Hyman and Shashkov to the case of unstructured polyhedral meshes. The accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem.
We propose a strategy for the systematic construction of the mimetic inner products on cochain spaces for the numerical approximation of partial differential equations on unstructured polygonal and ...polyhedral meshes. The mimetic inner products are locally built in a recursive way on each k-cell and, then, globally assembled. This strategy is similar to the implementation of the finite element methods. The effectiveness of this approach is documented by deriving mimetic discretizations and testing their behavior on a set of problems related to the Maxwell equations.
In the present paper we study a finite element method for the incompressible Stokes problem with a boundary immersed in the domain on which essential constraints are imposed. Such type of methods may ...be useful to tackle problems with complex geometries, interfaces such as multiphase flow and fluid–structure interaction. The method we study herein consists in locally refining elements crossed by the immersed boundary such that newly added elements, called subelements, fit the immersed boundary. In this sense, this approach is of a fitted type, but with an original mesh given independently of the location of the immersed boundary. We use such a subdivision technique to build a new finite element basis, which enables us to represent accurately the immersed boundary and to impose strongly Dirichlet boundary conditions on it. However, the subdivision process may imply the generation of anisotropic elements, which, for the incompressible Stokes problem, may result in the loss of inf–sup stability even for well-known stable element schemes. We therefore use a finite element approximation, which appears stable also on anisotropic elements. We perform numerical tests to check stability of the chosen finite elements. Several numerical experiments are finally presented to illustrate the capabilities of the method. The method is presented for two-dimensional problems.
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