Local Convergence of Adaptive Methods for Nonlinear Partial Differential Equations
Local Convergence of Adaptive Methods for Nonlinear Partial Differential Equations
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发表时间：
2010-01
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通讯作者：
M. Holst;G. Tsogtgerel;Yunrong Zhu
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作者：
M. Holst;G. Tsogtgerel;Yunrong Zhu
本文给出了Banach空间上抽象非线性算子方程的一类自适应逼近算法的收敛性理论，并利用该理论得到了几类非线性椭圆型方程的实用自适应有限元方法的收敛性结果.在本文的第一部分，我们建立了一个弱 * 收敛框架的非线性算子，其Gateaux导数是局部Lipschitz和满足局部inf-sup条件。该框架可以被视为延长最近的收敛结果的线性问题的莫林，Siebert和Veeser一般的非线性设置。在具有局部结构的Banach空间中，给出了非线性算子方程的一个抽象自适应逼近算法。然后将弱-* 收敛框架应用于这类抽象局部自适应算法，给出了一般收敛性结果。然后将收敛性结果应用于几个半线性和拟线性标量椭圆方程和椭圆方程组的标准AFEM算法，包括：具有亚临界非线性的半线性问题，稳定的Navier-Stokes方程和具有非线性扩散的拟线性问题.这产生了一些新的AFEM收敛结果，这些非线性问题。在本文的第二部分中，我们发展了第二个基于强压缩的抽象收敛框架，将Ca scon，Kreuzer，Nochetto和Siebert以及Mekchay和Nochetto最近关于线性问题的压缩结果推广到抽象非线性问题。然后，我们建立的条件下，它是可能的应用合同框架的抽象自适应算法定义的前面，给出了一个收缩的结果，适用于非线性问题的AFE型算法。然后将压缩结果应用于几个半线性标量椭圆方程的标准AFEM算法，包括：具有亚临界非线性的半线性问题，Poisson -Boltzmann方程和广义相对论中的Hamilton约束，在每种情况下产生AF EM压缩结果。
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 gen eral convergence result. The convergence result is then applied to a standard AFEM algorithm in the case of sev- eral semilinear and quasi-linear scalar elliptic equation s and elliptic systems, including: a semilinear problem with subcritical nonlinearity, the st eady 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, extend- ing the recent contraction results for linear problems of Ca scon, Kreuzer, Nochetto, and Siebert and of Mekchay and Nochetto to abstract nonlinear problems. We then establish conditions under which it is possible to apply the contracti on framework to the abstract adaptive algorithm defined earlier, giving a contraction re sult for AFEM-type algorithms applied to nonlinear problems. The contraction result is th en applied to a standard AFEM algorithm in the case of several semilinear scalar elliptic equations, including: a semi- linear problem with subcritical nonlinearity, the Poisson -Boltzmann equation, and the Hamiltonian constraint in general relativity, yielding AF EM contraction results in each case.