On the positivity of discrete harmonic functions and the discrete Harnack inequality for piecewise linear finite elements

Let D be a domain in Euclidean space of d dimensions and K a compact subset of D. The well known Harnack inequality assures the existence of a positive constant A depending only on D and K such that (l/A)u(x)<u(y)<Au(x) for all x and y in K and all positive harmonic functions u on D. In this we obtain a global integral version of this inequality under geometrical conditions on the domain. The result is the following: suppose D is a Lipschitz domain satisfying the uniform exterior sphere condition—stated in Section 2. If u is harmonic in D with continuous boundary data f thenwhere ds is the d — 1 dimensional Hausdorff measure on the boundary ժD. A large class of domains satisfy this condition. Examples are C2-domains, convex domains, etc.

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Brownian Motion and Harmonic Analysis on Sierpinski Carpets

Canadian Journal of Mathematics ◽

10.4153/cjm-1999-031-4 ◽

1999 ◽

Vol 51 (4) ◽

pp. 673-744 ◽

Cited By ~ 99

Author(s):

Martin T. Barlow ◽

Richard F. Bass

Keyword(s):

Brownian Motion ◽

Heat Equation ◽

Harmonic Analysis ◽

Heat Kernel ◽

Lower Bounds ◽

Harnack Inequality ◽

Harmonic Functions ◽

Upper And Lower Bounds ◽

Isotropic Diffusion ◽

Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

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International Workshop on Finite Elements for Microwave Engineering

10.36253/978-88-6655-968-9 ◽

2016 ◽

Keyword(s):

Finite Element ◽

Finite Elements ◽

Dominant Role ◽

Piecewise Linear ◽

International Workshop ◽

American Mathematical Society ◽

Element Formulation ◽

Microwave Engineering ◽

History Of ◽

Element Activity

When Courant prepared the text of his 1942 address to the American Mathematical Society for publication, he added a two-page Appendix to illustrate how the variational methods first described by Lord Rayleigh could be put to wider use in potential theory. Choosing piecewise-linear approximants on a set of triangles which he called elements, he dashed off a couple of two-dimensional examples and the finite element method was born. … Finite element activity in electrical engineering began in earnest about 1968-1969. A paper on waveguide analysis was published in Alta Frequenza in early 1969, giving the details of a finite element formulation of the classical hollow waveguide problem. It was followed by a rapid succession of papers on magnetic fields in saturable materials, dielectric loaded waveguides, and other well-known boundary value problems of electromagnetics. … In the decade of the eighties, finite element methods spread quickly. In several technical areas, they assumed a dominant role in field problems. P.P. Silvester, San Miniato (PI), Italy, 1992 Early in the nineties the International Workshop on Finite Elements for Microwave Engineering started. This volume contains the history of the Workshop and the Proceedings of the 13th edition, Florence (Italy), 2016 . The 14th Workshop will be in Cartagena (Colombia), 2018.

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Fast Domain Decomposition Type Solver for Stiffness Matrices of Reference p-Elements

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2013-0003 ◽

2013 ◽

Vol 13 (2) ◽

pp. 161-183 ◽

Cited By ~ 4

Author(s):

Vadim Korneev

Keyword(s):

Finite Difference ◽

Domain Decomposition ◽

Elliptic Equations ◽

Harmonic Functions ◽

Schur Complement ◽

P Elements ◽

Stiffness Matrices ◽

The Family ◽

Discrete Harmonic Functions

Abstract. A key component of domain decomposition solvers for hp discretizations of elliptic equations is the solver for internal stiffness matrices of p-elements. We consider an algorithm which belongs to the family of secondary domain decomposition solvers, based on the finite-difference like preconditioning of p-elements, and was outlined by the author earlier. We remove the uncertainty in the choice of the coarse (decomposition) grid solver and suggest the new interface Schur complement preconditioner. The latter essentially uses the boundary norm for discrete harmonic functions induced by orthotropic discretizations on slim rectangles, which was derived recently. We prove that the algorithm has linear arithmetical complexity.

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