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ON MAXIMUM NORM CONVERGENCE OF MULTIGRID METHODS FOR elliptic boundary-value-problems

SIAM JOURNAL ON NUMERICAL ANALYSIS 1994年第2期31卷 378-392页

作者： REUSKEN, A Eindhoven Univ of Technology Eindhoven Netherlands

Multigrid methods applied to standard linear finite element discretizations of linear elliptic boundary value problems in two dimensions,are considered. In the multigrid method, damped Jacobi or damped Gauss-Seidel is used as a smoother. It is proven that the two-grid method with nu pre-smoothing iterations has a contraction number with respect to the maximum norm that is (asymptotically) bounded by Cnu-1/2 \ln h(k)\2, with h(k);a suitable mesh size parameter, Moreover, it is shown that this bound is sharp in the sense that a factor \ln h(k)\ is necessary.

关键词： MULTIGRID CONVERGENCE ANALYSIS MAXIMUM NORM elliptic boundary value problems

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Analysis of the Diffuse Domain Method for Second Order elliptic boundary value problems

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FOUNDATIONS OF COMPUTATIONAL MATHEMATICS 2017年第3期17卷 627-674页

作者： Burger, Martin Elvetun, Ole Loseth Schlottbom, Matthias Univ Munster Inst Computat & Appl Math Einsteinstr 62 D-48149 Munster Germany Univ Munster Cells Mot Cluster Excellence Munster Germany Norwegian Univ Life Sci Dept Math Sci & Technol As Norway

The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper, we study the diffuse domain method for approximating second order elliptic boundary value problems posed on bounded domains and show convergence and rates of the approximations generated by the diffuse domain method to the solution of the original second order problem when complemented by Robin, Dirichlet or Neumann conditions. The main idea of the diffuse domain method is to relax these boundary conditions by introducing a family of phase-field functions such that the variational integrals of the original problem are replaced by a weighted average of integrals of perturbed domains. From a functional analytic point of view, the phase-field functions naturally lead to weighted Sobolev spaces for which we present trace and embedding results as well as various types of Poincar, inequalities with constants independent of the domain perturbations. Our convergence analysis is carried out in such spaces as well, but allows to draw conclusions also about unweighted norms applied to restrictions on the original domain. Our convergence results are supported by numerical examples.

关键词： Diffuse domain method Weighted Sobolev spaces Domain perturbations elliptic boundary value problems

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Multiple positive solutions of semilinear elliptic boundary value problems in the half space

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2012年第1期75卷 304-316页

作者： Hsu, Tsing-san Lin, Huei-li Chang Gung Univ Dept Nat Sci Ctr Gen Educ Tao Yuan 333 Taiwan

In this paper, we prove that if Q is a positive continuous function in R(N) and satisfies the suitable conditions, then for sufficiently small lambda > 0, -Delta u + u = Q(z)u(p-1) with the boundary condition u(x, z(N)) = lambda g(x) admits at least three positive solutions in R(+)(N). (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Multiple positive solutions elliptic boundary value problems The half space Category

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A QUASI-NEWTON METHOD FOR elliptic boundary-value-problems

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SIAM JOURNAL ON NUMERICAL ANALYSIS 1987年第3期24卷 516-531页

作者： KELLEY, CT SACHS, EW North Carolina State Univ Raleigh NC USA North Carolina State Univ Raleigh NC USA

For some boundary value problems for equations of the form $\nabla ^2 u + f(x,u,\nabla u) = 0$ with linear boundary conditions we give a convergence analysis for a quasi-Newton method that converges superlinearly in the $C^1 $ norm. This scheme for infinite-dimensional problems becomes a method for the finite-dimensional problems that are produced by discretizing the differential equation. This method is a generalization of a method proposed by Hart and Soul for discretizations of two point boundary value problems.

关键词： 49D15 65J15 quasi-Newton methods elliptic boundary value problems

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Multiplicity of solutions for a class of elliptic boundary value problems

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2009年第7-8期71卷 2606-2613页

作者： He, Xiaoming Zou, Wenming Cent Univ Nationalities Coll Sci Beijing 100081 Peoples R China Tsinghua Univ Dept Math Sci Beijing 100084 Peoples R China

In this paper we study the existence of infinitely many solutions for a class of elliptic boundary value problems. The existence results include both superlinear case and an asymptotically linear case. Our arguments are based on a recently given new Fountain Theorem due to Zou [W. Zou, Variant Fountain Theorems and their applications, Manuscripta Math. 104 (2001) 343-358]. (c) 2009 Elsevier Ltd. All rights reserved.

关键词： elliptic boundary value problems New fountain theorems Superlinear and asymptotically linear Minimax method

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Convergence rates for solving elliptic boundary value problems with singular parameterizations in isogeometric analysis

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COMPUTER AIDED GEOMETRIC DESIGN 2017年 52-53卷 170-189页

作者： Wu, Meng Wang, Yicao Mourrain, Bernard Nkonga, Boniface Cheng, Changzheng Hefei Univ Technol Sch Math Hefei Anhui Peoples R China Hohai Univ Dept Math Nanjing Jiangsu Peoples R China Inria Aromath Sophia Antipolis France Univ Nice Lab JA Dieudonne Nice France Hefei Univ Technol Sch Civil Engn Hefei Peoples R China

In this paper, we present convergence rates for solving elliptic boundary value problems with singular parameterizations in isogeometric analysis. First, the approximation errors with the L-2(Omega)-norm and the H-1(Omega)-seminorm are estimated locally. The impact of singularities is considered in this framework. Second, the convergence rates for solving PDEs with singular parameterizations are discussed. These results are based on a weak solution space that contains all of the weak solutions of elliptic boundary value problems with smooth coefficients. For the smooth weak solutions obtained by isogeometric analysis with singular parameterizations and the finite element method, both are shown to have the optimal convergence rates. For non-smooth weak solutions, the optimal convergence rates are reached by setting proper singularities of a controllable parameterization, even though convergence rates are not optimal by finite element method, and the convergence rates by isogeometric analysis with singular parameterizations are better than the ones by the finite element method. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Singular parameterizations Isogeometric analysis elliptic boundary value problems Approximation error estimates Convergence rates

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Convergence analysis of orthogonal spline collocation for elliptic boundary value problems

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SIAM JOURNAL ON NUMERICAL ANALYSIS 1998年第2期35卷 617-631页

作者： Bialecki, B Colorado Sch Mines Dept Math & Comp Sci Golden CO 80401 USA

Existence, uniqueness, and optimal order H-2, H-1, and L-2 error bounds are established for the orthogonal spline collocation solution of a Dirichlet boundary value problem on the unit square. The linear, elliptic, nonself-adjoint, partial differential equation is given in nondivergence form. The approximate solution, which is a tensor product of continuously differentiable piecewise polynomials of degree r greater than or equal to 3, is determined by satisfying the partial differential equation at the nodes of a composite Gauss quadrature.

关键词： elliptic boundary value problems piecewise polynomials Gauss points orthogonal spline collocation Sobolev spaces

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Besov regularity for elliptic boundary value problems in polygonal domains

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APPLIED MATHEMATICS LETTERS 1999年第6期12卷 31-36页

作者： Dahlke, S Rhein Westfal TH Aachen Inst Geometrie & Prakt Math D-52056 Aachen Germany

This paper is concerned with the regularity of the solutions to elliptic boundary value problems in polygonal domains Omega contained in R-2. Especially, we consider the specific scale B-tau(alpha)(L-tau(Omega)), 1/tau = alpha/2 + 1/p, of Besov spaces. The regularity of the variational solution in these Besov spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. The proofs are based on specific representations of the solutions which were, e.g., derived by Grisvard [1], and on characterizations of Besov spaces by wavelet expansions. (C) 1999 Elsevier Science Ltd. All rights reserved.

关键词： elliptic boundary value problems adaptive methods besov spaces wavelets

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The numerical methods for oscillating singularities in elliptic boundary value problems

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JOURNAL OF COMPUTATIONAL PHYSICS 2001年第2期170卷 742-763页

作者： Oh, HS Kim, H Lee, SJ Univ N Carolina Dept Math Charlotte NC 28223 USA Daebul Univ Dept Comp Sci Youngam South Korea Daejin Univ Dept Math Pochun South Korea

The singularities near the crack tips of homogeneous materials are monotone of type r(alpha) and r(alpha)log(delta)r, (depending on the boundary conditions along nonsmooth domains). However. the singularities around the interfacial cracks of the heterogeneous bimaterials are oscillatory of type, r(alpha) sin(epsilon logr). The method of auxiliary mapping (MAM). introduced by Babuska and Oh, was proven to be successful in dealing with, r(alpha) type singularities. However, the effectiveness of MAM is reduced in handling oscillating singularities. This paper deals with oscillating singularities as well as the monotone singularities by extending MAM through introducing the power auxiliary mapping and the exponential auxiliary mapping. (C) 2001 Academic Press.

关键词： monotone singularity oscillating singularity interfacial cracks elliptic boundary value problems the p-version (the h-p version) of finite element method method of auxiliary mapping the power (the exponential) auxiliary mapping

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Diagonalized Legendre spectral methods using Sobolev orthogonal polynomials for elliptic boundary value problems

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APPLIED NUMERICAL MATHEMATICS 2018年 127卷 196-210页

作者： Ai, Qing Li, Hui-yuan Wang, Zhong-qing Univ Shanghai Sci & Technol Sch Sci Shanghai 200093 Peoples R China Chinese Acad Sci Inst Software Lab Parallel Comp State Key Lab Comp Sci Beijing 100190 Peoples R China

Fully diagonalized spectral methods using Sobolev orthogonal/biorthogonal basis functions are proposed for solving second order elliptic boundary value problems. We first construct the Fourier-like Sobolev polynomials which are mutually orthogonal (resp. bi-orthogonal) with respect to the bilinear form of the symmetric (resp. unsymmetric) elliptic Neumann boundary value problems. The exact and approximation solutions are then expanded in an infinite and truncated series in the Sobolev orthogonal polynomials, respectively. An identity is also established for the a posterior error estimate with a simple error indicator. Further, the Fourier-like Sobolev orthogonal polynomials and the corresponding Legendre spectral method are proposed in parallel for Dirichlet boundary value problems. Numerical experiments illustrate that our Legendre methods proposed are not only efficient for solving elliptic problems but also equally applicable to indefinite Helmholtz equations and singular perturbation problems. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Legendre spectral method Sobolev orthogonal polynomials elliptic boundary value problems A posterior error estimates

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