Finite element exterior calculus with lower-order terms

Arnold, Douglas N.; Li, Lizao

doi:10.1090/mcom/3158

Cited by 7 publications

(6 citation statements)

References 18 publications

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“…Last, by means of the Bernoulli functions, the resulting schemes are shown to converge to special upwind schemes as the diffusion coefficient approaches to zero. The SAFE schemes also provide a promising way to discrete the Lie convection with Hodge Laplacian [2].The rest of the paper is organized as follows. In Section 2 we introduce the general convection-diffusion problems and briefly review the P − 1 Λ k discrete de Rham complex.…”

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confidence: 99%

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confidence: 99%

“…More specifically, we are motivated by the Edge-Averaged Finite Element (EAFE) method for scalar convection-diffusion problem proposed by Xu and Zikatanov [38]. There are two main advantages to using EAFE: (1) The monotonicity of EAFE can be established for a very general class of meshes; (2) The local stiffness matrix of EAFE can easily be obtained by modifying that of standard Poisson. A construction that ensures the general SPD diffusion coefficient matrix was proposed in [34].…”

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Simplex-Averaged Finite Element Methods for $H$(grad), $H$(curl), and $H$(div) Convection-Diffusion Problems

Wu¹,

Xu²

2020

SIAM J. Numer. Anal.

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This paper is devoted to the construction and analysis of the finite element approximations for the H(D) convection-diffusion problems, where D can be chosen as grad, curl or div in 3D case. An essential feature of these constructions is to properly average the PDE coefficients on the sub-simplexes. The schemes are of the class of exponential fitting methods that result in special upwind schemes when the diffusion coefficient approaches to zero. Their well-posedness are established for sufficiently small mesh size assuming that the convection-diffusion problems are uniquely solvable. Convergence of first order is derived under minimal smoothness of the solution. Some numerical examples are given to demonstrate the robustness and effectiveness for general convection-diffusion problems.The theory and numerical analysis of such a scalar convection-diffusion equation are well-studied in the literature. It is well known that, for small α, boundary layers may appear in the solution of (1.2) and standard finite element methods may suffer from strong numerical oscillations and instabilities if the mesh size is not small enough.Numerous studies on the stable discretization of scalar convection-diffusion have been published. In the finite element methods, various special strategies have been developed, including the stabilized discontinuous Galerkin method [33,10], SUPG method [12,22,14], bubble function stabilization [11,8,7,23], local projection stabilization [25], edge stabilization and continuous interior penalty method [16,13,15]. Other studies do not require the characteristics to be specified, such as exponential fitting [9,38,19,34,4] and Petrov-Galerkin method [35,18].The H(curl) and H(div) convection-diffusion problems have received more and more attention from numerical computation. The discretization of the general convection, known as extrusion, has been discussed via Whitney forms in [6]. For the pure advection problem, the stabilized Galerkin method has been extended from 0form [10] to 1-form [30] and k-form [28,32]. These discretizations of the advection problem, along with the proper discretization of the diffusion term, are feasible to tackle the general convection-diffusion problems. Besides the Eulerian method, the semi-Lagrangian method can be applied to the time-dependent convection-diffusion problems for differential forms [28,31,29].More specifically, we are motivated by the Edge-Averaged Finite Element (EAFE) method for scalar convection-diffusion problem proposed by Xu and Zikatanov [38]. There are two main advantages to using EAFE: (1) The monotonicity of EAFE can be established for a very general class of meshes; (2) The local stiffness matrix of EAFE can easily be obtained by modifying that of standard Poisson. A construction that ensures the general SPD diffusion coefficient matrix was proposed in [34]. A high-order Scharfetter-Gummel scheme, known as a high-order extension of EAFE, was given in [4]. Similar to (1.2), the standard finite element methods also seriously suffer from numerical ins...

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Simplex-Averaged Finite Element Methods for $H$(grad), $H$(curl), and $H$(div) Convection-Diffusion Problems

Wu¹,

Xu²

2020

SIAM J. Numer. Anal.

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“…Use of the IGAFEM to analyse a 3D solid presents challenges since it requires a volumetric NURBS description, which is not trivial to generate. There have been some early works injecting spline functions into the BEM, many of which predated the IGAFEM [12][13][14][15][16][17][18]. Simpson et al [19,20] applied the isogeometric concept in a BEM framework, and found similar benefits over classical piecewise polynomial formulations.…”

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Hybrid nearly singular integration for isogeometric boundary element analysis of coatings and other thin 2D structures

Gong

Trevelyan

Hattori

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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We present an isogeometric boundary element method (IGABEM) capable of delivering accurate and efficient solutions in the heat transfer analysis of 2D coated structures such as those commonly found in turbomachinery. Although we consider very thin coatings (of thickness down to 10 −7 m), they are modelled explicitly as BEM zones, and this is made possible by the development of a new integration scheme (sinh +) aimed particularly at the challenging nearly singular integrals that arise. Sinh + is a hybrid of adaptive and sinh transformation approaches, and we make further extensions to the latter to improve its robustness. The scheme is tuned to deliver results of engineering accuracy in an optimal time. The scheme is adaptable, by changing a tolerance, to enable engineers to achieve a different balance between accuracy and computational efficiency as may be required for different applications. A set of numerical examples demonstrates the ability of the scheme to produce accurate temperature distributions efficiently in the presence of very thin coatings.

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“…In view of literature on traditional mixed methods and MFEMs in FEEC, researchers are more concerned with errors measured in the L 2 -norm · or the V -norm · HΛ . For example, the authors in [3,4] made a great effort to obtain improved decoupled a priori error estimates for σ − σ h and u − u h . The contributions of this paper are listed as follows.…”

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Some Convergence and Optimality Results of Adaptive Mixed Methods in Finite Element Exterior Calculus

Li¹

2019

SIAM J. Numer. Anal.

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In this paper, we present several new a posteriori error estimators and two adaptive mixed finite element methods AMFEM1 and AMFEM2 for the Hodge Laplacian problem in finite element exterior calculus. We prove that AMFEM1 and AMFEM2 are both convergent starting from any initial coarse mesh. In addition, we prove the quasi-optimality of AMFEM2. Comparing to existing literature, our results work on Lipschitz domains with nontrivial cohomology and provide the first norm convergence and quasi-optimality results for the Hodge Laplacian. Key words. a posteriori error estimate, adaptive mixed finite element method, finite element exterior calculus, Hodge Laplacian, convergence, optimality AMS subject classifications. 65N12, 65N15, 65N30, 65N50, 41A25 ∇ − → H(curl, Ω) curl −−→ H(div, Ω) div − − → L 2 (Ω).

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Finite element exterior calculus with lower-order terms

Cited by 7 publications

References 18 publications

Simplex-Averaged Finite Element Methods for $H$(grad), $H$(curl), and $H$(div) Convection-Diffusion Problems

Simplex-Averaged Finite Element Methods for $H$(grad), $H$(curl), and $H$(div) Convection-Diffusion Problems

Hybrid nearly singular integration for isogeometric boundary element analysis of coatings and other thin 2D structures

Some Convergence and Optimality Results of Adaptive Mixed Methods in Finite Element Exterior Calculus

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