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    Finite element approximation of partial differential equations
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    Apr 1987 - Sep 2002
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    National Research Council
    Pavia, Lombardy, Italy
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    Peter Wriggers
    Sudeep Kundu
    Francesca Gardini
    Steffen Weißer
    Sheng-Da Zeng
    Mateo Ríos
    Michał Bosy
    Sergii V. Siryk
    Xiangyun Meng
    Benjamin Sourcis
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    Cristina Tablino-Possio
    Gianmarco Manzini
    Franco Brezzi
    Alejandro Ortiz-Bernardin
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    We aim to develop new numerical methods to solve partial differential equations (mainly elliptic and parabolic) on unstructured meshes of cells with very general geometric shapes. These methods may be considered in the major classes of numerical discretizations like Finite Volumes and Finite Elements, as DDFV, MFD, VEM, PFEM, GS, etc.
    Project
    Enhance the robustness of meshfree Galerkin methods for solid mechanics simulations by designing improved numerical integration schemes on arbitrary integration cells that result in consistent, stable and optimally convergent solutions.
    Project
    In this paper we initiate the investigation of Virtual Elements with curved faces. We consider the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain or with a curved interface. While an approximation of the domain with polygons leads, for $k \geq 2$, to a sub-optimal rate of convergence, we show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence.
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    Research Items (68)
    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.
    The authors study the use of the virtual element method (VEM for short) of order k for general second order elliptic problems with variable coefficients in three space dimensions. Moreover, they investigate numerically also the serendipity version of the VEM and the associated computational gain in terms of degrees of freedom.
    In this paper we initiate the investigation of Virtual Elements with curved faces. We consider the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain or with a curved interface. While an approximation of the domain with polygons leads, for k ≥ 2, to a sub-optimal rate of convergence, we show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence.
    We numerically validate the Virtual Element Method of order k for general second order elliptic problems with variable coefficients in three dimensions. Moreover, we investigate numerically also the Serendipity version of the VEM (in three dimensions) and the associated computational gain in terms of degrees of freedom.
    This paper summarizes the development of Veamy, an object-oriented C++ library for the virtual element method (VEM) on general polygonal meshes, whose modular design is focused on its extensibility. The two-dimensional linear elastostatic problem has been chosen as the starting stage for the development of this library. The theory of the VEM in which Veamy is based upon is presented using a notation and a terminology that resemble the language of the finite element method in engineering analysis. Several examples are provided to demonstrate the usage of Veamy, and in particular, one of them features the interaction between Veamy and the polygonal mesh generator PolyMesher. Veamy is free and open source software.
    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.
    We develop a numerical assessment of the Virtual Element Method for the discretization of a diffusion-reaction model problem, for higher "polynomial" order k and three space dimensions. Although the main focus of the present study is to illustrate some h-convergence tests for different orders k, we also hint on other interesting aspects such as structured polyhedral Voronoi meshing, robustness in the presence of irregular grids, sensibility to the stabilization parameter and convergence with respect to the order k.
    Over the past two decades, meshfree methods have undergone significant development as a numerical tool to solve partial differential equations (PDEs). In contrast to finite elements, the basis functions in meshfree methods are smooth (nonpolynomial functions), and they do not rely on an underlying mesh structure for their construction. These features render meshfree methods to be particularly appealing for higher-order PDEs and for large deformation simulations of solid continua. However, a deficiency that still persists in meshfree Galerkin methods is the inaccuracies in numerical integration, which affects the consistency and stability of the method. Several previous contributions have tackled the issue of integration errors with an eye on consistency, but without explicitly ensuring stability. In this paper, we draw on the recently proposed virtual element method, to present a formulation that guarantees both the consistency and stability of the approximate bilinear form. We adopt maximum-entropy meshfree basis functions, but other meshfree basis functions can also be used within this framework. Numerical results for several two- and three-dimensional elliptic (Poisson and linear elastostatic) boundary-value problems that demonstrate the effectiveness of the proposed formulation are presented.
    In the present work, we analyze the $hp$ version of Virtual Element methods for the 2D Poisson problem. We prove exponential convergence of the energy error employing sequences of polygonal meshes geometrically refined, thus extending the classical choices for the decomposition in the $hp$ Finite Element framework to very general decomposition of the domain. A new stabilization for the discrete bilinear form with explicit bounds in $h$ and $p$ is introduced. Numerical experiments validate the theoretical results. We also exhibit a numerical comparison between $hp$ Virtual Elements and $hp$ Finite Elements.
    In the present paper we detail the implementation of the Virtual Element Method for two dimensional elliptic equations in primal and mixed form with variable coefficients.
    We analyse the Virtual Element Methods (VEM) on a simple elliptic model problem, allowing for more general meshes than the one typically considered in the VEM literature. For instance, meshes with arbitrarily small edges (with respect to the parent element diameter), can be dealt with. Our general approach applies to different choices of the stability form, including, for example, the "classical" one introduced 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), no. 1, 199-214], and a recent one presented in [Wriggers, P., Rust, W.T., and Reddy, B.D., A virtual element method for contact, submitted for publication]. Finally, we show that the stabilization term can be simplified by dropping the contribution of the internal-to-the-element degrees of freedom. The resulting stabilization form, involving only the boundary degrees of freedom, can be used in the VEM scheme without affecting the stability and convergence properties. The numerical tests are in accordance with the theoretical predictions.
    We extend the basic idea of Serendipity Virtual Elements from the previous case (by the same authors) of nodal ($H^1$-conforming) elements, to a more general framework. Then we apply the general strategy to the case of $H(div)$ and $H(curl)$ conforming Virtual Element Methods, in two and three dimensions.
    The Virtual Element Method (in short VEM) is a recent generalization of the Finite Element Method that can easily handle general polygonal and polyhedral meshes. In this short note we will present three variants of the Virtual Element Method, the only difference being the number of internal degrees of freedom. We will see that all methods behave in a very similar way.
    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.
    In the present paper we initiate the study of $hp$ Virtual Elements. We focus on the case with uniform polynomial degree across the mesh and derive theoretical convergence estimates that are explicit both in the mesh size $h$ and in the polynomial degree $p$ in the case of finite Sobolev regularity. Exponential convergence is proved in the case of analytic solutions. The theoretical convergence results are validated in numerical experiments. Finally, an initial study on the possible choice of local basis functions is included.
    In the present paper we construct virtual element spaces that are \(H(\mathrm{div})\)-conforming and \(H(\mathbf{curl})\)-conforming on general polygonal and polyhedral elements; these spaces can be interpreted as a generalization of well known finite elements. We moreover present the basic tools needed to make use of these spaces in the approximation of partial differential equations. Finally, we discuss the construction of exact sequences of VEM spaces.
    We introduce and analyze a virtual element method (VEM) for the Helmholtz problem with approximating spaces made of products of low order VEM functions and plane waves. We restrict ourselves to the 2D Helmholtz equation with impedance boundary conditions on the whole domain boundary. The main ingredients of the plane wave VEM scheme are: i) a low frequency space made of VEM functions, whose basis functions are not explicitly computed in the element interiors; ii) a proper local projection operator onto the high-frequency space, made of plane waves; iii) an approximate stabilization term. A convergence result for the h-version of the method is proved, and numerical results testing its performance on general polygonal meshes are presented.
    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. Beirão 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 (Formula presented.)-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. Beirão 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. Beirão 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.
    In this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C0 Lagrange finite element methods. In the VEM, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of -continuous VEMs on polygonal and polyhedral meshes are described, which reveals that the variational approach adopted in the VEM affords a generalized and robust means to stabilize underintegrated finite elements. We focus on the heat conduction (Poisson) equation and present a virtual element approach for the isoparametric four-node quadrilateral and eight-node hexahedral elements. In addition, we show quantitative comparisons of the consistency and stabilization matrices in the VEM with those in the hourglass control method of Belytschko and coworkers. Numerical examples in two and three dimensions are presented for different stabilization parameters, which reveals that the method satisfies the patch test and delivers optimal rates of convergence in the L2 norm and the H1 seminorm for Poisson problems on quadrilateral, hexahedral, and arbitrary polygonal meshes. Copyright © 2015 John Wiley & Sons, Ltd.
    In the present paper we construct Virtual Element Spaces that are $H({\rm div})$-conforming and $H({\rm \bf curl})$-conforming on general polygonal and polyhedral elements; these spaces can be interpreted as a generalization of well known Finite Elements. We moreover present the basic tools needed to make use of these spaces in the approximation of partial differential equations. Finally, we discuss the construction of exact sequences of VEM spaces.
    The paper is devoted to the spectral analysis of effective preconditioners for linear systems obtained via a finite element approximation to diffusion-dominated convection–diffusion equations. We consider a model setting in which the structured finite element partition is made by equilateral triangles. Under such assumptions, if the problem is coercive and the diffusive and convective coefficients are regular enough, then the proposed preconditioned matrix sequences exhibit a strong eigenvalue clustering at unity, the preconditioning matrix sequence and the original matrix sequence are spectrally equivalent, and under the constant coefficients assumption, the eigenvector matrices have a mild conditioning. The obtained results allow to prove the conjugate gradient optimality and the generalized minimal residual quasi-optimality in the case of structured uniform meshes. The interest of such a study relies on the observation that automatic grid generators tend to construct equilateral triangles when the mesh is fine enough. Numerical tests, both on the model setting and in the non-structured case, show the effectiveness of the proposal and the correctness of the theoretical findings. Copyright © 2014 John Wiley & Sons, Ltd.
    Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finite difference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary polygonal meshes was devised. The method was coined as the virtual element method (VEM), since it did not require the explicit construction of basis functions. This advance provides a more in-depth understanding of mimetic schemes, and also endows polygonal-based Galerkin methods with greater flexibility than three-node and four-node finite element methods. In the VEM, a projection operator is used to realize the decomposition of the stiffness matrix into two terms: a consistent matrix that is known, and a stability matrix that must be positive semi-definite and which is only required to scale like the consistent matrix. In this paper, we first present an overview of previous developments on conforming polygonal and polyhedral finite elements, and then appeal to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinate-based Galerkin method on polygonal and polyhedral elements. The consistent matrix of the VEM is adopted, and numerical quadrature with generalized barycentric coordinates is used to compute the stability matrix. This facilitates post-processing of field variables and visualization in the VEM, and on the other hand, provides a means to exactly satisfy the patch test with efficient numerical integration in polygonal and polyhedral finite elements. We present numerical examples that demonstrate the sound accuracy and performance of the proposed method. For Poisson problems in ℝ2 and ℝ3, we establish that linearly complete generalized barycentric interpolants deliver optimal rates of convergence in the L2-norm and the H1-seminorm.
    We present the essential ingredients in the Virtual Element Method for a simple linear elliptic second-order problem. We emphasize its computer implementation, which will enable interested readers to readily implement the method.
    The paper is devoted to the spectral analysis of effective preconditioners for linear systems obtained via a Finite Element approximation to diffusion-dominated convection-diffusion equations. We consider a model setting in which the structured finite element partition is made by equi-lateral triangles. Under such assumptions, if the problem is coercive, and the diffusive and convective coefficients are regular enough, then the proposed preconditioned matrix sequences exhibit a strong clustering at unity, the preconditioning matrix sequence and the original matrix sequence are spectrally equivalent, and the eigenvector matrices have a mild conditioning. The obtained results allow to show the optimality of the related preconditioned Krylov methods. %It is important to stress that The interest of such a study relies on the observation that automatic grid generators tend to construct equi-lateral triangles when the mesh is fine enough. Numerical tests, both on the model setting and in the non-structured case, show the effectiveness of the proposal and the correctness of the theoretical findings.
    In the original virtual element space with degree of accuracy k, projector operators in the H^1-seminorm onto polynomials of degree @?k can be easily computed. On the other hand, projections in the L^2 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 L^2 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 L^2 error estimates.
    We present, on the simplest possible case, what we consider as the very basic features of the (brand new) virtual element method. As the readers will easily recognize, the virtual element method could easily be regarded as the ultimate evolution of the mimetic finite differences approach. However, in their last step they became so close to the traditional finite elements that we decided to use a different perspective and a different name. Now the virtual element spaces are just like the usual finite element spaces with the addition of suitable non-polynomial functions. This is far from being a new idea. See for instance the very early approach of E. Wachspress [A Rational Finite Element Basic (Academic Press, 1975)] or the more recent overview of T.-P. Fries and T. Belytschko [The extended/generalized finite element method: An overview of the method and its applications, Int. J. Numer. Methods Engrg.84 (2010) 253–304]. The novelty here is to take the spaces and the degrees of freedom in such a way that the elementary stiffness matrix can be computed without actually computing these non-polynomial functions, but just using the degrees of freedom. In doing that we can easily deal with complicated element geometries and/or higher-order continuity conditions (like C1, C2, etc.). The idea is quite general, and could be applied to a number of different situations and problems. Here however we want to be as clear as possible, and to present the simplest possible case that still gives the flavor of the whole idea.
    An advection-diffusion equation is considered, for which the solution is advection-dominated in most of the domain. A domain decomposition method based on a self-adaptive, smooth coupling of the reduced advection equation and the full advection-diffusion equation is proposed. The convergence of an iteration-by-subdomain method is investigated.
    In this paper we discuss a way to recover a classical residual-based error estimator for elliptic problems by using a finite element space enriched with bubble functions. The advection-dominated case is also discussed.
    In this paper we analyze the Residual-Free Bubble (RFB) method applied to the linear diffusion–advection–reaction problem. We propose a new a priori error analysis for the method and for its practical implementation in a quite general context, which allows, e.g. linear or quadratic elements on the resolvable scales. We also perform some numerical tests, showing in both cases the advantages of the method.
    In this paper we consider a nonlinear modification of a linear convection-diffusion problem in order to get a pure convection equation where the original problem is convection dominated. We extend the results of previous papers by considering mixed Dirichlet/Oblique derivative boundary conditions.
    We propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms. Our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes. The a priori error analysis relies on the connection between the mimetic formulation and the lowest order Raviart-Thomas mixed finite element method. The theoretical results are confirmed by numerical experiments.
    A two-step preconditioned iterative method based on the Hermitian and skew-Hermitian splitting is applied to the solution of nonsymmetric linear systems arising from the finite element approximation of diffusion-dominated convection-diffusion equations. The theoretical spectral analysis focuses on the case of matrix sequences related to finite element approximations on uniform structured meshes, by referring to spectral tools derived from Toeplitz theory. In such a setting, if the problem is coercive and the diffusive and convective coefficients are regular enough, then the proposed preconditioned matrix sequence shows a strong clustering at unity, i.e., a superlinear preconditioning sequence is obtained. Under the same assumptions, the optimality of the preconditioned Hermitian and skew-Hermitian splitting (PHSS) method is proved, and some numerical experiments confirm the theoretical results. Tests on unstructured meshes are also presented, showing the same convergence behavior.
    A two-step preconditioned iterative method based on the Hermitian/Skew-Hermitian splitting is applied to the solution of nonsymmetric linear systems arising from the Finite Element approximation of convection-diffusion equations. The theoretical spectral analysis focuses on the case of matrix sequences related to FE approximations on uniform structured meshes, by referring to spectral tools derived from Toeplitz theory. In such a setting, if the problem is coercive, and the diffusive and convective coefficients are regular enough, then the proposed preconditioned matrix sequence shows a strong clustering at unity, i.e., a superlinear preconditioning sequence is obtained. Under the same assumptions, the optimality of the PHSS method is proved and some numerical experiments confirm the theoretical results. Tests on unstructured meshes are also presented, showing the some convergence behavior.
    We present a finite volume method for the numerical approximation of advection–diffusion problems in convection-dominated regimes. The method works on unstructured grids formed by convex polygons of any shape and yields a piecewise linear approximation to the exact solution which is second-order accurate away from boundary and internal layers. Basically, we define a constant approximation of the solution gradient in every mesh cell which is expressed by using the cell averages of the solution within the adjacent cells. A careful design of the reconstruction algorithm for cell gradients and the introduction in the discrete formulation of a special non-linear term, which plays the role of the artificial diffusion, allows the method to achieve shock-capturing capability. We emphasize that no slope limiters are required by this approach. Optimal convergence rates, as theoretically expected, and non-oscillatory behavior close to layers are confirmed by numerical experiments.
    In this paper we present an RGB to XYZ transformation method based on a pattern search optimization algorithm. Whatever strategy is adopted to initialize the color transformation, our method is able to optimize it in order to minimize the color error on a given training set, also taking into account a color mapping constraint. We report experimental results on simulated and real data sets showing that our method significantly outperforms existing ones.
    The purpose of this paper is to present the residual-free bubbles (RFB) method and to show its relationship with the streamline-upwind Petrov/Galerkin method (SUPG). The paper is organized as follows. After the Introduction, we show how the failure of classical finite element methods for convection-dominated convection–diffusion problem can be explained by energy considerations. Then we describe the residual-free bubbles ideas, and in the last section we show the equivalence of RFB and SUPG in a simple case.
    SUPG and residual-free bubbles are closely related methods that have been used with success to stabilize a certain number of problems, including advection-dominated flows. In recent times, a slightly different idea has been proposed: to choose a suitable subgrid in each element, and then solving Standard Galerkin on the Augmented Grid. For this, however, the correct location of the subgrid node(s) plays a crucial role. Here, for the model problem of linear advection–diffusion equations, we propose a simple criterion to choose a single internal node such that the corresponding plain-Galerkin scheme on the augmented grid provides the same a priori error estimates that are typically obtained with SUPG or RFB methods.
    We introduce a Galerkin formulation for the advective-reactive-diffusive equation. It is based on “residual-free bubble” enrichments for the test and trial spaces. An approximation of the ideal residual-free bubbles is considered and a new stabilized method is derived. The resulting formulation is proven to be stable for a wide range of coefficients and a convergence estimate is established. Numerical experiments attest to the stability and accuracy of the approach introduced.
    We show that, when solving a linear system with an iterative method, it is necessary to measure the error in the space in which the residual lies. We present examples of linear systems which emanate from the finite element discretization of elliptic partial differential equations, and we show that, when we measure the residual in H −1(Ω), we obtain a true evaluation of the error in the solution, whereas the measure of the same residual with an algebraic norm can give misleading information about the convergence. We also state a theorem of functional compatibility that proves the existence of perturbations such that the approximate solution of a PDE is the exact solution of the same PDE perturbed.
    In this article, we analyze a discontinuous finite element method recently introduced by Bassi and Rebay for the approximation of elliptic problems. Stability and error estimates in various norms are proven. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 365–378, 2000
    We consider conforming Petrov–Galerkin formulations for the advective and advective–diffusive equations. For the linear hyperbolic equation, the continuous formulation is set up using different spaces and the discretization follows with different “bubble” enrichments for the test and trial spaces. Boundary conditions for residual-free bubbles are modified to accommodate with the first-order equation case and regular bubbles are used to enrich the other space. Using piecewise linears with these enrichments, the final formulations are shown to be equivalent to the SUPG method, provided the data are assumed to be piecewise constant. Generalization to include diffusion is also presented.
    The standard Galerkin method can be roughly described as being an approximation of the variational formulation of a PDE (or system of PDE’s) in a space of functions that is spanned by piecewise polynomials. This simple idea presents several advantages: first, the discrete system of equations that arise from such an approximation is going to be “banded” since the piecewise polynomials can be constructed to have a “small” support, and therefore the matrices involved are sparse. Second, taking derivatives and integrating polynomials is a very attractive task for any first year calculus student, and the simplicity of the implementation of the method for the most cumbersome PDE or system of PDE’s seems straightforward. Third, the mathematical analysis seems to be possible without a lot of sophistication (at least if we have an elliptic problem, and we disregard technicalities referring to domain shape, etc.).
    We consider conforming Petrov-Galerkin formulations for the advective and advective -diffusive equations. For the linear hyperbolic equation, the continuous formulation is set up using different spaces and the discretization follows with different "bubble" enrichments for the test and trial spaces. Boundary conditions for residual-free bubbles are modified to accommodate with the first order equation case and regular bubbles are used to enrich the other space. Using piecewise linears with these enrichments, the final formulations are shown to be equivalent to the SUPG method, provided the data is assumed to be piecewise constant. Generalization to include diffusion is also presented. 1 Introduction The linear hyperbolic equation is simply the "pure" advective equation, where we drop the diffusion term from an advection-diffusion equation. The immediate consequence is that the equation is first order and therefore suitable boundary conditions have to be considered to make the problem w...
    . We develop an a priori error analysis of a nite element approximation to the elliptic advection-dioeusion equation Gamma"Deltau + a Delta ru = f subject to a homogeneous Dirichlet boundary condition, based on the use of residual-free bubble functions. An optimal order error bound is derived in the so-called stability-norm / "krvk 2 L 2(OmegaGamma + X T h T ka Delta rvk 2 L 2 (T ) ! 1=2 ; where h T denotes the diameter of element T in the subdivision of the computational domain. 1. Introduction. Suppose thatOmega is a bounded polygonal domain in the plane and assume, for simplicity, that a = (a 1 ; a 2 ) is a two-component vector function whose entries are constant onOmega . Assume further that f is a piecewise constant function dened onOmega . We note that our results are valid under more general hypotheses on the data (which will be discussed in the nal section) and in any number of space dimensions. Given that " is a positive constant, we consider the elliptic ...
    this paper, we present a new way for stabilizing Dirichlet problems with Lagrange multipliers for the particular case where u is approximated by a piecewise linear continuous function, and the Lagrange multipliers are approximated by piecewise constant functions on a nonmatching grid. Our stabilization is made by adding suitable bubble functions only on the triangles having an edge on the boundary. It is interesting to note that elimination of the bubbles by static condensation leads to a scheme very similar to that introduced a long time ago by Nitsche [DW95] and recently reproposed and analyzed in [Osw95]. For the sake of simplicity, we shall only discuss a single-domain problem. The extension to many subdomains can then be carried out by means of the usual coupling procedures (Dirichlet-Dirichlet or Neumann-Neumann or something else). The organization of the paper is the following. In Sect. 2 we present the singledomain problem, where the Dirichlet condition is imposed via Lagrange multipliers. In Sect. 3 we discuss its discretization with nonmatching grids and the bubble stabilization. In Sect. 4 we show that it is possible to eliminate both bubbles and Lagrange multipliers, thus obtaining a scheme that is easy to implementation and that strongly resembles the one discussed in [DW95, Osw95]. If needed, the Lagrange multipliers can be recovered by a simple and economical post-processing. This will be useful in a true domain decomposition situation, in order to carry out the iterative procedure. 2 The Single Domain Problem In order to introduce our stabilization technique we shall consider a problem on a single domain, thinking of it as one of the subdomains. Always referring for simplicity to the global problem (1.1), at each step of the domain decomposition procedu...
    We study a stabilized spectral method for the incompressible Navier-Stokes equations; stabilization is achieved by using the residual-free bubbles approach. Numerical results for the regularized driven cavity and the backward-facing step are presented.
    We further consider the Galerkin method for advective-diffusive equations in two dimensions. The finite dimensional space employed is of piecewise polynomials enriched with residual-free bubbles (RFB). We show that, in general, this method does not coincide with the SUPG method, unless the piecewise polynomials are spanned by linear functions. Furthermore, a simple stability analysis argument displays the effect of the RFB on the reduced space of piecewise polynomials, which, in some situations, is not equivalent to streamline diffusion for bilinears.
    An overview of the unusual stabilized finite element method and of the standard Galerkin method enriched with residual free bubble functions is presented. For the first method a concrete model problem illustrates its application in advective-diffusive-reactive equations and for the second method it is shown how static condensation of residual free bubbles gives rise to mass lumping and selective reduced integration, which are viewed as numerical tricks and can now be derived by the standard Galerkin method without tricks.
    Residual-free bubbles have been recently introduced in order to compute optimal values for the stabilization methods `a la Hughes-Franca. However, unless in very special situations, (one-dimensional problems, limit cases, etc.) they require the actual solution of PDE problems (the bubble problems) in each element. Thus they are very difficult to be used in practice. In this paper we present, for the special case of convection-dominated elliptic problems, a cheap way to compute approximately the solution of the bubble problem in each element. This provides, as a consequence, a cheap way to compute good approximations for the optimal values of the stabilization parameters. 1 Introduction We will present in this paper a new stabilization method for convection-diffusion problems, particularly designed to treat strongly convection-dominated problems, but able to adapt naturally from diffusion-dominated regime to convection-dominated regime in a very simple way. We will consider, for the sa...
    Linear-constant velocity-pressure elements are enriched with residualfree macro bubbles. Static condensation prompts a stabilized method for this element, where mesh-dependent jumps of the normal stress are added across internal edges of the underlying macroelements. This procedure renders the SIMPLEST element stable. 1 Introduction The Stokes problem models creeping flows, which are flows where inertia can be neglected with respect to diffusion and source terms in the momentum equilibrium equation. Applications of this model to some subsurface flows, using variable viscosity coefficients, has gained new impetus with the increased concern in environmental conservation issues. In addition to these applications, the Stokes operator is a major component of the Navier-Stokes equations, and therefore a building block to understanding the different flow regimes in a more complex fluid dynamics problem. Approximation of the Stokes problem using finite element methods produces a mixed variati...
    . We show that three well-known "variational crimes" in finite elements -- upwinding, mass lumping and selective reduced integration -- may be derived from the Galerkin method employing the standard polynomial-based finite element spaces enriched with residual-free bubbles. 1. Introduction. In this note we introduce a finite element method based on enriching the classical polynomial-based finite element spaces with residual-free bubbles. We show that in 1D the classical techniques of upwinding, mass lumping and selective reduced integration can be derived by the Galerkin method based on the enriched space. 2. An abstract presentation. LetOmega ae R n be a regular domain, f 2 L 2(Omega ) and (2.1) ( Lu = f inOmega u = 0 on @Omega be a linear elliptic boundary value problem which can be given a classical variational formulation as follows: (2.2) find u 2 V such that a (u; v) = F (v) for all v 2 V where a (1; 1) is a bilinear form on V = H 1 0(Omega0 which is conti...
    Residual-free bubbles are derived for the Timoshenko beam problem. Eliminating these bubbles the resulting formulation is form-identical to using the following tricks to the standard variational formulation: i) one-point reduced integration on the shear energy term; ii) replace its coefficient 1=ffl 2 by 1=(ffl 2 + (h 2 K =12)) in each element; iii) modify consistently the right-hand-side. This final formulation is `legally' obtained in that the Galerkin method enriched with residual-free bubbles is developed using full integration throughout. Furthermore this method is nodally exact by construction. Submitted to: Computer Methods in Applied Mechanics and Engineering Preprint July 1995 i L.P. Franca and A. Russo Preprint, July 1995 1 1. INTRODUCTION The deflection of a beam taking into account bending and shear deformations is described by the Timoshenko model. Standard Galerkin finite element method using equal-order piecewise linear approximations for the unknown dependent...
    In this paper we show the equivalence between the variational multiscale and the residual-free bubbles concepts. 1 Introduction The subject of stabilized finite element methods has existed now for over sixteen years. The basic reason for introducing stabilized methods is that straightforward application of the Galerkin version of the finite element method to certain problems of mathematical physics and engineering yields numerical approximations that are deficient in that they do not inherit the stability properties of the continuous problem. Stabilized methods constitute a systematic methodology for improving stability behavior without compromising accuracy. As such, stabilized methods have provided fundamental solutions to the problem of discrete approximations in several practically important areas. Perhaps the most notable area is fluid dynamics. Convection-diffusion operators have perplexed numerical analysts for decades. Historically they have been treated by methods which hav...
    In this paper we show the equivalence between the variational multiscale and the residual-free bubbles concepts.
    We show that three well-known “variational crimes” in finite elements—upwinding, mass lumping and selective reduced integration—may be derived from the Galerkin method employing the standard polynomial-based finite element spaces enriched with residual-free bubbles.
    We present an overview of stabilized finite element methods and of the standard Galerkin method enriched with residual-free bubble functions. The inadequacy of the standard Galerkin method using piecewise polynomials is discussed for different applications; the treatment using stabilized methods in their different versions is reviewed; and the connection to the standard Galerkin method with richer subspaces follows using the subgrid method or the residual-free-bubbles viewpoint. We close with a discussion on how to approximate the exact problem suggested by residual-free bubbles. 1 The standard Galerkin method and some of its failures The standard Galerkin method can be roughly described as being an approximation of the variational formulation of a PDE (or system of PDE's) in a space of functions that is spanned by piecewise polynomials. This simple idea presents several advantages: first, the discrete system of equations that arise from such an approximation is going to be "banded" s...
    . A nodally exact scheme is derived for a model equation in 1D involving zeroth and second order terms. The method is derived using residual-free bubbles in conjunction with the Galerkin approximation. It is shown that this approach leads to the mass lumping scheme for sufficiently small mesh sizes. 1. The residual-free bubbles approach. The usage of the Galerkin method enriched with bubble functions has gained new impetus recently [1], [2]. In particular, the observation that this approach gives rise to streamline upwinding [2], [3], finally brought together two apparently distinct discretization procedures: namely, the employment of the standard Galerkin method with, possibly, more complex functions and the practice of combining the Galerkin method with least-squares like terms, viewed as upwind schemes. An abstract theory has been put together constructing a bridge between the stabilized methods and the Galerkin method using standard polynomial finite elements plus "virtual bubbles...
    . We show that three well-known "variational crimes" in finite elements -- upwinding, mass lumping and selective reduced integration -- may be derived from the Galerkin method employing the standard polynomial-based finite element spaces enriched with residual-free bubbles. 1. Introduction. In this note we introduce a finite element method based on enriching the classical polynomial-based finite element spaces with residual-free bubbles. We show that in 1D the classical techniques of upwinding, mass lumping and selective reduced integration can be derived by the Galerkin method based on the enriched space. 2. An abstract presentation. LetOmega ae R n be a regular domain, f 2 L 2(Omega ) and (2.1) ( Lu = f inOmega u = 0 on @Omega be a linear elliptic boundary value problem which can be given a classical variational formulation as follows: (2.2) find u 2 V such that a (u; v) = F (v) for all v 2 V where a (1; 1) is a bilinear form on V = H 1 0(Omega0 which is conti...
    In this paper we discuss the stabilization, via bubble functions, of a finite element method for stationary linearized incompressible Navier-Stokes equations. It is shown that ‘residual free’ bubbles can reproduce SUPG.
    A stable finite element scheme for advection dominated problems is presented. The method is based on a classical piecewise linear continuous approximation of the solution and is proved to verify the discrete maximum principle whenever the triangulation is of weakly acute type. Several numerical tests confirm robustness of the method.
    A residual-based error estimator for the MINI-element approximation of the Stokes problem is considered. It is shown that the bubble part of the MINI-element approximation is a term of this estimator.
    The purpose of this paper is to give a simple uniqueness result for a class of nonlinear partial differential equations. As a particular case, we obtain the uniqueness of the solution for the problem studied in [1], where this question had been left open.
    An abstract is not available.
    In this paper we employ equivariant singularity theory to study the postbuckling behavior of a cylindrical shell under axial compression, obtaining some results about the existence of secondary bifurcations and how they are connected to each other. The basic idea, first employed by Bauer, Keller and Reiss in [1], and then coupled with singularity theory by Schaeffer and Golubitsky in [16] and [17] and by Buzano in [4], consists in unfolding a multiple eigenvalue, obtained by forcing two eigenvalues to coalesce by varying the geometric parameters of the shell. This approah is made possible by a general analysis of bifurcation problems invariant with respect to the symmetries of the cylinder i.e. with respect to the group O(2)Z 2.
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