Finite element quasi-interpolation and best approximation
01 Jul 2017-Mathematical Modelling and Numerical Analysis (EDP Sciences)-Vol. 51, Iss: 4, pp 1367-1385
TL;DR: In this paper, a quasi-interpolation operator for scalar and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces is introduced.
Abstract: This paper introduces a quasi-interpolation operator for scalar-and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces. This operator is stable in L^1 , is a projection, whether homogeneous boundary conditions are imposed or not, and, assuming regularity in the fractional Sobolev spaces W^{s,p} where p ∈ [1, ∞] and s can be arbitrarily close to zero, gives optimal local approximation estimates in any L^p-norm. The theory is illustrated on H^1-, H(curl)-and H(div)-conforming spaces.
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11 Aug 2018
TL;DR: The core properties that are required to prove the convergence of a GDM are stressed, and the analysis of the method is performed on a series of elliptic and parabolic problems.
Abstract: This monograph is dedicated to the presentation of the Gradient Discretisation Method (GDM) and of some of its applications. It is intended for masters students, researchers and experts in the field of the numerical analysis of partial differential equations.
The GDM is a framework which contains classical and recent discretisation schemes for diffusion problems of different kinds: linear or non linear, steady-state or time-dependent. The schemes may be conforming or non conforming and may rely on very general polygonal or polyhedral meshes. In this monograph, the core properties that are required to prove the convergence of a GDM are stressed, and the analysis of the method is performed on a series of elliptic and parabolic problems, linear or non-linear, for which the GDM is particularly adapted. As a result, for these models, any scheme entering the GDM framework is known to converge. A key feature of this monograph is the presentation of techniques and results which enable a complete convergence analysis of the GDM on fully non-linear, and sometimes degenerate, models. The scope of some of these techniques and results goes beyond the GDM, and makes them potentially applicable to numerical schemes not (yet) known to fit into this framework.
Appropriate tools are then developed so as to easily check whether a given scheme satisfies the expected properties of a GDM. Thanks to these tools a number of methods can be shown to enter the GDM framework: some of these methods are classical, such as the conforming Finite Elements, the Raviart–Thomas Mixed Finite Elements, or the P 1 non-conforming Finite Elements. Others are more recent, such as the Hybrid Mixed Mimetic or Nodal Mimetic methods, some Discrete Duality Finite Volume schemes, and some Multi-Point Flux Approximation schemes.
134 citations
TL;DR: An h p -Clement type quasi-interpolation operator is obtained, i.e., an operator that requires minimal smoothness of the function to be approximated but has the expected approximation properties in terms of the local mesh size and polynomial degree.
Abstract: We develop a regularization operator based on smoothing on a spatially varying length scale. This operator is defined for functions u ? L 1 and has approximation properties that are given by the local Sobolev regularity of u and the local smoothing length scale. Additionally, the regularized function satisfies inverse estimates commensurate with the approximation orders. By combining this operator with a classical h p -interpolation operator, we obtain an h p -Clement type quasi-interpolation operator, i.e., an operator that requires minimal smoothness of the function to be approximated but has the expected approximation properties in terms of the local mesh size and polynomial degree. As a second application, we consider residual error estimates in h p -boundary element methods that are explicit in the local mesh size and the local approximation order.
104 citations
TL;DR: A class ofumerical homogenization methods that are very closely related to the method of M{\aa}lqvist and Peterseim, but do not make explicit or implicit use of a scale separation.
Abstract: Numerical homogenization tries to approximate the solutions of elliptic partial differential equations with strongly oscillating coefficients by functions from modified finite element spaces. We present in this paper a class of such methods that are very closely related to the method of M{\aa}lqvist and Peterseim [Math. Comp. 83, 2014]. Like the method of M{\aa}lqvist and Peterseim, these methods do not make explicit or implicit use of a scale separation. Their compared to that in the work of M{\aa}lqvist and Peterseim strongly simplified analysis is based on a reformulation of their method in terms of variational multiscale methods and on the theory of iterative methods, more precisely, of additive Schwarz or subspace decomposition methods.
87 citations
TL;DR: This contribution extends the idea of modification only in the right-hand side of a Stokes discretization to low and high order Taylor--Hood and mini elements, which have continuous discrete pressures.
Abstract: Classical inf-sup stable mixed finite elements for the incompressible (Navier--)Stokes equations are not pressure-robust, i.e., their velocity errors depend on the continuous pressure. However, a modification only in the right-hand side of a Stokes discretization is able to reestablish pressure-robustness, as shown recently for several inf-sup stable Stokes elements with discontinuous discrete pressures. In this contribution, this idea is extended to low and high order Taylor--Hood and mini elements, which have continuous discrete pressures. For the modification of the right-hand side a velocity reconstruction operator is constructed that maps discretely divergence-free test functions to exactly divergence-free ones. The reconstruction is based on local $H(\mathrm{div})$-conforming flux equilibration on vertex patches, and fulfills certain orthogonality properties to provide consistency and optimal a priori error estimates. Numerical examples for the incompressible Stokes and Navier--Stokes equations conf...
77 citations
TL;DR: A Hybrid High-Order method on unfitted meshes to approximate elliptic interface problems and proves stability estimates and optimal error estimates in the $H^1$-norm.
Abstract: We design and analyze a Hybrid High-Order (HHO) method on unfitted meshes to approximate elliptic interface problems. The curved interface can cut through the mesh cells in a very general fashion. As in classical HHO methods, the present unfitted method introduces cell and face unknowns in uncut cells, but doubles the unknowns in the cut cells and on the cut faces. The main difference with classical HHO methods is that a Nitsche-type formulation is used to devise the local reconstruction operator. As in classical HHO methods, cell unknowns can be eliminated locally leading to a global problem coupling only the face unknowns by means of a compact stencil. We prove stability estimates and optimal error estimates in the H 1-norm. Robustness with respect to cuts is achieved by a local cell-agglomeration procedure taking full advantage of the fact that HHO methods support polyhedral meshes. Robustness with respect to the contrast in the material properties from both sides of the interface is achieved by using material-dependent weights in Nitsche's formulation.
58 citations
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
Book•
01 Apr 1985
TL;DR: Second-order boundary value problems in polygons have been studied in this article for convex domains, where the second order boundary value problem can be solved in the Sobolev spaces of Holder functions.
Abstract: Foreword Preface 1. Sobolev spaces 2. Regular second-order elliptic boundary value problems 3. Second-order elliptic boundary value problems in convex domains 4. Second-order boundary value problems in polygons 5. More singular solutions 6. Results in spaces of Holder functions 7. A model fourth-order problem 8. Miscellaneous Bibliography Index.
5,248 citations
01 Jan 2008
TL;DR: In this paper, a variational method in the theory of harmonic integrals has been proposed to solve the -Neumann problem on strongly pseudo-convex manifolds and parametric Integrals two-dimensional problems.
Abstract: Semi-classical results.- The spaces Hmp and Hmp0.- Existence theorems.- Differentiability of weak solutions.- Regularity theorems for the solutions of general elliptic systems and boundary value problems.- A variational method in the theory of harmonic integrals.- The -Neumann problem on strongly pseudo-convex manifolds.- to parametric Integrals two dimensional problems.- The higher dimensional plateau problems.
3,190 citations