Multilevel correction adaptive finite element method for semilinear elliptic equation

Lin, Qun; Xie, Hehu; Xu, Fei

doi:10.1007/s10492-015-0110-x

Cited by 18 publications

(7 citation statements)

References 16 publications

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“…In recent years, numerical methods for Steklov eigenvalue problems have attracted more and more scholars' attention (see, e.g., [3,4,5,6,9,13,21,26,27,28,31,36,40]). It is well known that in the numerical approximation of partial differential equations, the adaptive procedures based on a posteriori error estimates, due to the less computational cost and time, are the mainstream direction and have gained an enormous importance.…”

Section: Introductionmentioning

confidence: 99%

An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem

Yang

2016

Applied Numerical Mathematics

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This paper proposes and analyzes an a posteriori error estimator for the finite element multi-scale discretization approximation of the Steklov eigenvalue problem. Based on the a posteriori error estimates, an adaptive algorithm of shifted inverse iteration type is designed. Finally, numerical experiments comparing the performances of three kinds of different adaptive algorithms are provided, which illustrate the efficiency of the adaptive algorithm proposed here.Keywords : Steklov eigenvalue problem, finite element, multi-scale discretization, adaptive algorithm, a posteriori error estimate.

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Section: Introductionmentioning

confidence: 99%

An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem

Yang

2016

Applied Numerical Mathematics

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“…Thus, the adaptive refinement is adopted to couple with the full multigrid method described in Algorithm 3.2 (cf. [13]).…”

Section: Examplementioning

confidence: 99%

A Full Multigrid Method For Semilinear Elliptic Equation

Xie¹,

Xu²

2016

Preprint

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A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.

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“…Recently, two-level [62,40] and multilevel [37,38,39,60,61] correction methods have been proposed to reduce the complexity of solving eigenpairs associated with low eigenvalues by first solving a coarse mesh/scale approximation, which can then be corrected by solving linear systems (corresponding to linearized eigenvalue problems) on a hierarchy of finer meshes/scales. Although the multilevel correction approach has been extended to multigrid methods for linear and nonlinear eigenvalue problems [18,37,38,39,27,60,61], the regularity estimates required for linear complexity do not hold for PDEs with rough coefficients and a naive application of the correction approach to multiscale eigenvalue problems may converge very slowly. For two-level methods [62] this lack of robustness can be alleviated by numerical homogenization techniques [40], e.g., the so-called Localized Orthogonal Decomposition (LOD) method.…”

Section: Introductionmentioning

confidence: 99%

Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction

Xie¹,

Zhang²,

Owhadi³

2019

SIAM J. Numer. Anal.

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We present a method for the fast computation of the eigenpairs of a bijective positive symmetric linear operator L. The method is based on a combination of operator adapted wavelets (gamblets) with hierarchical subspace correction. First, gamblets provide a raw but fast approximation of the eigensubspaces of L by block-diagonalizing L into sparse and well-conditioned blocks. Next, the hierarchical subspace correction method, computes the eigenpairs associated with the Galerkin restriction of L to a coarse (low dimensional) gamblet subspace, and then, corrects those eigenpairs by solving a hierarchy of linear problems in the finer gamblet subspaces (from coarse to fine, using multigrid iteration). The proposed algorithm is robust to the presence of multiple (a continuum of) scales and is shown to be of near-linear complexity when L is an (arbitrary local, e.g. differential) operator mapping H s 0 (Ω) to H −s (Ω) (e.g. an elliptic PDE with rough coefficients).

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Multilevel correction adaptive finite element method for semilinear elliptic equation

Cited by 18 publications

References 16 publications

An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem

An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem

A Full Multigrid Method For Semilinear Elliptic Equation

Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction

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