Journal ArticleDOI
Discrete Hodge operators
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TLDR
A generic discrete Hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations, reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus.Abstract:
Many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms. Discrete differential forms provide a natural and canonical approach to their discretization. However, much freedom remains concerning the choice of discrete Hodge operators, that is, discrete analogues of constitutive laws. A generic discrete Hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations. We reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus.read more
Citations
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Journal ArticleDOI
Finite elements in computational electromagnetism
TL;DR: In this paper, finite element Galerkin schemes for a number of linear model problems in electromagnetism were discussed, and the finite element schemes were introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms.
Journal ArticleDOI
Discrete exterior calculus
TL;DR: In this article, a theory of discrete exterior calculus (DEC) is proposed, which includes not only discrete equivalents of differential forms, but also discrete vector fields and the operators acting on these objects.
Journal ArticleDOI
Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces
Ralf Hiptmair,Jinchao Xu +1 more
TL;DR: This paper develops and analyzes a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of H(curl, )- and H(div,)-elliptic variational problems and proves mesh-independent effectivity of the precondITIONers by using the abstract theory of auxiliary space preconditionsing.
Journal ArticleDOI
Virtual Element Methods for plate bending problems
TL;DR: In this article, the authors discuss the application of virtual elements to linear plate bending problems, in the Kirchhoff-love formulation, and show that the treatment of the C 1 -continuity condition is much easier than for traditional finite elements.
Book ChapterDOI
Principles of Mimetic Discretizations of Differential Operators
Pavel B. Bochev,James M. Hyman +1 more
TL;DR: This work provides a common framework for mimetic discretizations using algebraic topology to guide the analysis and demonstrates how to apply the framework for compatible discretization for two scalar versions of the Hodge Laplacian.
References
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Journal ArticleDOI
Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media
Abstract: Maxwell's equations are replaced by a set of finite difference equations. It is shown that if one chooses the field points appropriately, the set of finite difference equations is applicable for a boundary condition involving perfectly conducting surfaces. An example is given of the scattering of an electromagnetic pulse by a perfectly conducting cylinder.
Book
Finite Element Method for Elliptic Problems
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.
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The Mathematical Theory of Finite Element Methods
TL;DR: In this article, the construction of a finite element of space in Sobolev spaces has been studied in the context of operator-interpolation theory in n-dimensional variational problems.
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Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms
TL;DR: This paper presents the results of an analysis of the "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions and its applications to Mixed Approximation and Homogeneous Stokes Equations.