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Author Holst, Michael ♦ Tsogtgerel, Gantumur ♦ Zhu, Yunrong
Source CiteSeerX
Content type Text
File Format PDF
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Adaptive Method ♦ Nonlinear Partial Differential Equation ♦ Local Convergence ♦ Linear Problem ♦ Nonlinear Problem ♦ Convergence Result ♦ Semilinear Problem ♦ Standard Afem Algorithm ♦ Banach Space ♦ Adaptive Algorithm ♦ Contraction Result ♦ Weak Convergence Framework ♦ Subcritical Nonlinearity ♦ Second Part ♦ First Part ♦ Several Semilinear Scalar Elliptic Equation ♦ General Class ♦ Several Class ♦ Practical Adaptive Finite Element Method ♦ Convergence Theory ♦ Recent Contraction Result ♦ Strong Contraction ♦ Nonlinear Elliptic Equation ♦ Gateau Derivative ♦ Local Inf-sup Condition ♦ General Relativity ♦ Steady Navier-stokes Equation ♦ Poisson-boltzmann Equation ♦ General Nonlinear Setting ♦ General Convergence Result ♦ Local Structure ♦ Recent Convergence Result ♦ Hamiltonian Constraint ♦ Afem-type Algorithm ♦ Nonlinear Diffusion ♦ Afem Contraction Result ♦ Several New Afem Convergence Result ♦ Abstract Adaptive Approximation Algorithm ♦ Adaptive Approximation Algorithm ♦ Nonlinear Operator Equation ♦ Quasilinear Problem ♦ Quasi-linear Scalar Elliptic Equation ♦ Second Abstract Convergence Framework ♦ Nonlinear Operator ♦ Elliptic System ♦ Abstract Nonlinear Operator Equation
Abstract ABSTRACT. In this article we develop convergence theory for a general class of adaptive approximation algorithms for abstract nonlinear operator equations on Banach spaces, and then use the theory to obtain convergence results for practical adaptive finite element methods (AFEM) applied to several classes of nonlinear elliptic equations. In the first part of the paper, we develop a weak- * convergence framework for nonlinear operators, whose Gateaux derivatives are locally Lipschitz and satisfy a local inf-sup condition. The framework can be viewed as extending the recent convergence results for linear problems of Morin, Siebert and Veeser to a general nonlinear setting. We formulate an abstract adaptive approximation algorithm for nonlinear operator equations in Banach spaces with local structure. The weak- * convergence framework is then applied to this class of abstract locally adaptive algorithms, giving a general convergence result. The convergence result is then applied to a standard AFEM algorithm in the case of several semilinear and quasi-linear scalar elliptic equations and elliptic systems, including: a semilinear problem with subcritical nonlinearity, the steady Navier-Stokes equations, and a quasilinear problem with nonlinear diffusion. This yields several new AFEM convergence results for these nonlinear problems. In the second part of the paper we develop a second abstract convergence framework based on strong contraction, extending the recent contraction results for linear problems of Cascon, Kreuzer, Nochetto, and Siebert and of Mekchay and Nochetto to abstract nonlinear problems. We then establish adaptive algorithm defined earlier, giving a contraction result for AFEM-type algorithms applied to nonlinear problems. The contraction result is then applied to a standard AFEM algorithm in the case of several semilinear scalar elliptic equations, including: a semilinear problem with subcritical nonlinearity, the Poisson-Boltzmann equation, and the Hamiltonian constraint in general relativity, yielding AFEM contraction results in each case.
Educational Role Student ♦ Teacher
Age Range above 22 year
Educational Use Research
Education Level UG and PG ♦ Career/Technical Study
Learning Resource Type Article