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32
Finite element exterior calculus, homological techniques, and applications
 ACTA NUMERICA
, 2006
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FINITE ELEMENT EXTERIOR CALCULUS: FROM HODGE THEORY TO NUMERICAL STABILITY
, 2009
"... Abstract. This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to diff ..."
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Cited by 90 (5 self)
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Abstract. This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the wellposedness of the continuous problem. The discretization methods we consider are finite element methods, in which a variational or weak formulation of the PDE problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace. After Hilbert space framework for analyzing the stability and convergence of such discretizations. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures that ensure wellposedness of the continuous problem from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they can be arranged into a subcomplex of this Hilbert complex, and there exists a bounded cochain projection from that complex to the subcomplex. In the next part of the paper, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially the elasticity complex and its application to the equations of elasticity. Background material is included to make the presentation selfcontained for a variety of readers.
R.: Smoothed projections in finite element exterior calculus
 Math. Comp
, 2008
"... Abstract. The development of smoothed projections, constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator, have proved to be an effective tool in finite element exterior calculus. The advantage of these operators is that they are ..."
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Cited by 30 (5 self)
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Abstract. The development of smoothed projections, constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator, have proved to be an effective tool in finite element exterior calculus. The advantage of these operators is that they are L2 bounded projections, and still they commute with the exterior derivative. In the present paper we generalize the construction of these smoothed projections, such that also non quasi–uniform meshes and essential boundary conditions are covered. The new tool introduced here is a space dependent smoothing operator which commutes with the exterior derivative. 1.
EQUILIBRATED RESIDUAL ERROR ESTIMATOR FOR EDGE ELEMENTS
"... Abstract. Reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem. A cheap computation of such functions via equilibration is well known for scalar equations of second order. We simpl ..."
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Cited by 27 (1 self)
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Abstract. Reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem. A cheap computation of such functions via equilibration is well known for scalar equations of second order. We simplify and modify the equilibration such that it can be applied to the curlcurl equation and edge elements. The construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions. For this reason, Raviart–Thomas elements are extended in the spirit of distributions. 1.
Equilibrated residual error estimators for Maxwell’s equations
 Johann Radon Institute for Computational and Applied Mathematics (RICAM
, 2006
"... Abstract. A posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem. A cheap computation of such functions via equilibration is wellknown for scalar equations of second order. We simplify and ..."
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Cited by 11 (2 self)
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Abstract. A posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem. A cheap computation of such functions via equilibration is wellknown for scalar equations of second order. We simplify and modify the equilibration such that it can be applied to Maxwell’s equations and edge elements. The construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions. For this reason, Raviart–Thomas elements are extended in the spirit of distributions. 1.
Positivity of certain sums over Jacobi Kernel Polynomials
, 2008
"... We present a computerassisted proof of positivity of sums over kernel polynomials for ultraspherical Jacobi polynomials. ..."
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Cited by 7 (3 self)
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We present a computerassisted proof of positivity of sums over kernel polynomials for ultraspherical Jacobi polynomials.
Convergence of adaptive edge element methods for the 3D eddy currents equations
 J. Comp. Math
"... Abstract. We consider an Adaptive Edge Finite Element Method (AEFEM) for the 3D eddy currents equations with variable coefficients using a residualtype a posteriori error estimator. Both the components of the estimator and certain oscillation terms, due to the occurrence of the variable coefficient ..."
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Cited by 6 (3 self)
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Abstract. We consider an Adaptive Edge Finite Element Method (AEFEM) for the 3D eddy currents equations with variable coefficients using a residualtype a posteriori error estimator. Both the components of the estimator and certain oscillation terms, due to the occurrence of the variable coefficients, have to be controlled properly within the adaptive loop which is taken care of by appropriate bulk criteria. Convergence of the AEFEM in terms of reductions of the energy norm of the discretization error and of the oscillations is shown. Numerical results are given to illustrate the performance of the AEFEM. Key words. adaptive edge elements, 3D eddy currents equations, convergence analysis, error and oscillation reduction, residual type a posteriori error estimates AMS subject classifications. 65F10, 65N30 1. Introduction. The
A posteriori error estimates for finite element exterior calculus: The de rham complex. arXiv:1203.0803v3
, 2012
"... Abstract. Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk and Winther [4] inclu ..."
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Cited by 4 (0 self)
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Abstract. Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk and Winther [4] includes a welldeveloped theory of finite element methods for Hodge Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of residual type for ArnoldFalkWinther mixed finite element methods for Hodgede Rham Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell’s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by unified treatment of the various Hodge Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge Laplacian. 1.
A mixed method for the biharmonic problem based on a system of firstorder equations
 SIAM J. Numer. Anal
"... Abstract. We introduce a new mixed method for the biharmonic problem. The method is based on a formulation where the biharmonic problem is rewritten as a system of four firstorder equations. A hybrid form of the method is introduced which allows to reduce the globally coupled degrees of freedom to ..."
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Abstract. We introduce a new mixed method for the biharmonic problem. The method is based on a formulation where the biharmonic problem is rewritten as a system of four firstorder equations. A hybrid form of the method is introduced which allows to reduce the globally coupled degrees of freedom to only those associated with Lagrange multipliers which approximate the solution and its derivative at the faces of the triangulation. For k ≥ 1 a projection of the primal variable error superconverges with order k+3 while the error itself converges with order k+ 1 only. This fact is exploited by using local postprocessing techniques that produce new approximations to the primal variable converging with order k + 3. We provide numerical experiments that validate our theoretical results.