Journal ArticleDOI
On the conditioning of finite element equations with highly refined meshes
Randolph E. Bank,L. R. Scott +1 more
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It is proven that the condition number of the linear system representing a finite element discretization of an elliptic boundary value problem does not degrade significantly as the mesh is refined locally, provided the mesh remains nondegenerate and a natural scaling of the basis functions is used.Abstract:
It is proven that the condition number of the linear system representing a finite element discretization of an elliptic boundary value problem does not degrade significantly as the mesh is refined locally, provided the mesh remains nondegenerate and a natural scaling of the basis functions is used. Bounds for the Euclidean condition number as a function of the number of degrees f freedom are derived in $n \geq 2$ dimensions. When $n \geq 3$ the bound is the same as for the regular mesh case, but when $n = 2$ a factor appears in the bound for the condition number that is logarithmic in the ratio of the maximum and minimum mesh sizes. Applications of the results to the conjugate-gradient iterative method for solving such linear systems are given.read more
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Delaunay Mesh Generation
TL;DR: The authors present algorithms for generating high-quality meshes in polygonal and polyhedral domains and illustrate how to use restricted Delaunay triangulations to extend the algorithms to surfaces with ridges and patches and volumes with smooth surfaces.
Proceedings ArticleDOI
Isosurface stuffing: fast tetrahedral meshes with good dihedral angles
TL;DR: The isosurface stuffing algorithm is the first algorithm that rigorously guarantees the suitability of tetrahedra for finite element methods in domains whose shapes are substantially more challenging than boxes.
Journal ArticleDOI
Preconditioned quantum linear system algorithm.
TL;DR: A quantum algorithm that generalizes the quantum linear system algorithm to arbitrary problem specifications is described and it is shown how it can be used to compute the electromagnetic scattering cross section of an arbitrary target exponentially faster than the best classical algorithm.
Book ChapterDOI
Aggressive Tetrahedral Mesh Improvement
TL;DR: This work introduces a topological transformation that inserts a new vertex (sometimes deleting others at the same time) and describes a schedule for applying and composing these operations that rarely gets stuck in a bad optimum.
References
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Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
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Singular Integrals and Differentiability Properties of Functions.
TL;DR: Stein's seminal work Real Analysis as mentioned in this paper is considered the most influential mathematics text in the last thirty-five years and has been widely used as a reference for many applications in the field of analysis.
Book
The Finite Element Method for Elliptic Problems
Philippe G. Ciarlet,J. T. Oden +1 more
TL;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.
Book
Analysis of Numerical Methods
TL;DR: Reference Record created on 2005-11-18, modified on 2016-08-08 as discussed by the authors, created on 2011-11 -18, created on 2006-11 18 and modified on 2008-08 -08,08