AFEM.bib

Babuska.I;Vogelius.M1984

ABSTRACT: This paper examines the concepts of feedback and adaptivity for the Finite Element Method. The model problem concernsC 0 elements of arbitrary, fixed degree for a one-dimensional two-point boundary value problem. Three different feedback methods are introduced and a detailed analysis of their adaptivity is given.

Bansch.E;Morin.P;Nochetto.R2002

ABSTRACT: We introduce and study an adaptive finite element method (FEM) for the Stokes system based on an Uzawa outer iteration to update the pressure and an elliptic adaptive inner iteration for velocity. We show linear convergence in terms of the outer iteration counter for the pairs of spaces consisting of continuous finite elements of degree k for velocity, whereas for pressure the elements can be either discontinuous of degree k-1 or continuous of degree k-1 and k. The popular Taylor--Hood family is the sole example of stable elements included in the theory, which in turn relies on the stability of the continuous problem and thus makes no use of the discrete inf-sup condition. We discuss the realization and complexity of the elliptic adaptive inner solver and provide consistent computational evidence that the resulting meshes are quasi-optimal.

Binev.P;Dahmen.W;DeVore.R2002a

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Binev.P;Dahmen.W;DeVore.R2004

ABSTRACT: Adaptive Finite Element Methods for numerically solving ellip- tic equations are used often in practice. Only recently [12], [17] have these methods been shown to converge. However, this convergence analysis says nothing about the rates of convergence of these methods and therefore does, in principle, not guarantee yet any numerical advantages of adaptive strat- egies versus non-adaptive strategies. The present paper modifies the adap- tive method of Morin, Nochetto, and Siebert [17] for solving the Laplace equation with piecewise linear elements on domains in R2 by adding a coars- ening step and proves that this new method has certain optimal convergence rates in the energy norm (which is equivalent to the H 1 norm). Namely, it is shown that whenever s > 0 and the solution u is such that for each n 1, it can be approximated to accuracy O(ns ) in the energy norm by a continu- ous, piecewise linear function on a triangulation with n cells (using complete knowledge of u), then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only infor- mation gained during the computational process. Moreover, the number of arithmetic computations in the proposed method is also of order O(n) for each n 1. The construction and analysis of this adaptive method relies on the theory of nonlinear approximation.

Binev.P;DeVore.R2004

ABSTRACT: Adaptive methods of approximation arise in many settings includ- ing numerical methods for PDEs and image processing. They can usually be described by a tree which records the adaptive decisions. This paper is concerned with the fast computation of near optimal trees based on n adap- tive decisions. The best tree based on n adaptive decisions could be found by examining all such possibilities. However, this is exponential in n and could be numerically prohibitive. The main result of this paper is to show that it is possible to find near optimal trees using computations linear in n.

Carstensen.C;Hoppe.R2005

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Carstensen.C;Hoppe.R2005a

ABSTRACT: An adaptive nonconforming finite element method is developed and analyzed that provides an error reduction due to the refinement process and thus guarantees convergence of the noncon- forming finite element approximations. The analysis is carried out for the lowest order Crouzeix-Raviart elements and leads to the linear convergence of an appropriate adaptive nonconforming fi- nite element algorithm with respect to the number of refinement levels. Important tools in the convergence proof are a discrete local efficiency and a quasi-orthogonality property. The proof does nei- ther require regularity of the solution nor uses duality arguments. As a consequence on the data control, no particular mesh design has to be monitored.

Carstensen.C;Hoppe.R2005b

ABSTRACT: For the 2D eddy currents equations, we design an adaptive edge finite element method (AEFEM) that guarantees an error reduction of the global discretization error in the H (curl)-norm and thus establishes convergence of the adaptive scheme. The error reduction property relies on a residual-type a posteriori error estimator and is proved for discretizations based on the lowest order edge elements of Nédélec's first family. The main ingredients of the proof are the reliability and the strict discrete local efficiency of the estimator as well as the Galerkin orthogonality of the edge element approximation.

Carstensen.C;Orlando.A;Valdman.J2005

ABSTRACT: The boundary value problem representing one time step of the pri- mal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some non-differentiable functional. This paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and, up to higher order terms, the R linear convergence of the stresses with respect to the number of loops. Applica- tions include several plasticity models: linear isotropic-kinematic hardening, linear kinematic hardening, and multisurface plasticity as model for nonlinear hardening laws. For perfect plasticity the adaptive algorithm yields strong convergence of the stresses. Numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule.

Carstensen.C2006

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Chen.L;Holst.M;Xu.J2006

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Dorfler.W;Wilderotter.O2000

ABSTRACT: We study the adaptive finite element method to solve linear elliptic boundary value problems on bounded domains in 2. For this we first prove a posteriori error estimates that carefully take data error into account and show convergence of an adaptive algorithm. Then we propose an adaptive method that may start from very coarse meshes. A numerical example underlines the necessity of monitoring the data error in applications. Moreover, we can show that the a posteriori error bound of our proposed error estimator will (in a simple model situation) not depend on jumps in the coefficient of the main part of the equation when the lines of discontinuity are resolved by the mesh.

Dorfler.W1996

ABSTRACT: We construct a converging adaptive algorithm for linear elements applied to Poisson's equation in two space dimensions. Starting from a macro triangulation, we describe how to construct an initial triangulation from a priori information. Then we use a posteriod error estimators to get a sequence of refined triangulations and approximate solutions. It is proved that the error, measured in the energy norm, decreases at a constant rate in each step until a prescribed error bound is reached. Extensions to higher-order elements in two space dimensions and numerical results are included.

Kondratyuk.Y2006

ABSTRACT: Although adaptive finite element methods (FEMs) are recognized as power- ful techniques for solving mixed variational problems of fluid mechanics, usually they are not even proven to converge. Only recently, in [SINUM, 40 (2002), pp.1207-1229] B ?ansch, Morin and Nochetto introduced an adaptive Uzawa FEM for solving the Stokes problem, and showed its convergence. In their paper, numerical experiments indicate (quasi-) opti- mal triangulations for some values of the parameters, where, a theoretical explanation of these results is still open. In this paper, we present a similar adaptive Uzawa finite element algorithm that uses a generalization of the optimal adaptive FEM of Stevenson [SINUM, 42 (2005), pp.2188- 2217] as an inner solver. By adding a derefinement step to the resulting adaptive Uzawa algorithm, in order to optimize the underlying triangulation after each fixed number of iterations, we show that the overall method converges with optimal rates with linear computational complexity.

Mekchay.K;Nochetto.R2005

ABSTRACT: We prove convergence of adaptive finite element methods (AFEMs) for general (nonsymmetric) second order linear elliptic PDEs, thereby extending the result of Morin, Nochetto, and Siebert [{\it SIAM J. Numer.\ Anal.}, 38 (2000), pp. 466--488; {\it SIAM Rev.}, 44 (2002), pp. 631--658]. The proof relies on quasi-orthogonality, which accounts for the bilinear form not being a scalar product, together with novel error and oscillation reduction estimates, which now do not decouple. We show that AFEMs are a contraction for the sum of energy error plus oscillation. Numerical experiments, including oscillatory coefficients and {both coercive and noncoercive} convection-diffusion PDE, illustrate the theory and yield optimal meshes.

Morin.P;Nochetto.R;Siebert.K2000

ABSTRACT: Data oscillation is intrinsic information missed by the averaging process associated with finite element methods (FEM) regardless of quadrature. Ensuring a reduction rate of data oscil- lation, together with an error reduction based on a posteriori error estimators, we construct a simple and efficient adaptive FEM for elliptic partial differential equations (PDEs) with linear rate of conver- gence without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance.

Morin.P;Nochetto.R;Siebert.K2002

ABSTRACT: Adaptive finite element methods (FEMs) have been widely used in applications for over 20 years now. In practice,they converge starting from coarse grids,although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators,together with a reduction rate of data oscillation (information missed by the underlying averaging process),we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

Nochetto.R2006

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Stevenson.R2005a

ABSTRACT: In this paper, an adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity. Whenever for some s > 0, the solution can be approximated within a tol- erance > 0 in energy norm by a continuous piecewise linear function on some partition with O( 1/s ) triangles, and one knows how to approximate the right-hand side in the dual norm with the same rate with piecewise constants, then the adaptive method produces approximations that con- verge with this rate, taking a number of operations that is of the order of the number of triangles in the output partition. The method is similar in spirit to that from [SINUM, 38 (2000), pp. 466--488] by Morin, Nochetto, and Siebert, and so in particular it does not rely on a recurrent coarsening of the partitions. Although the Poisson equation in two dimensions with piecewise linear approximation is considered, the results generalize in sev- eral respects.

Stevenson.R2006

ABSTRACT: Recently, in [Ste05], we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations, which is essentially the method from [SINUM, 38 (2000), 466--488] by Morin, Nochetto, and Siebert, converges with the optimal rate. The number of triangles N in the output partition of such a method is generally larger than the number M of triangles that in all intermediate partitions have been marked for bisection, because additional bisections are needed to retain conforming meshes. A key ingredient to our proof was a result from [Numer. Math., 97(2004), 219--268] by Binev, Dahmen and DeVore saying that N N0 C M for some absolute constant C , where N 0 is the number of triangles from the initial partition that have never been bisected. In this paper, we extend this result to bisection algorithms of n-simplices, with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions.

Veeser.A2002

ABSTRACT: The numerical solution of the homogeneous Dirichlet problem for the p-Laplacian is considered. We propose an adaptive algorithm with continuous piecewise affine finite elements and prove that the approximate solutions converge to the exact one. While the algorithm is a rather straight-forward generalization of those for the linear case p = 2, the proof of its convergence is different. In particular, it does not rely on a strict error reduction.

Wu.H;Chen.Z2003

ABSTRACT: In this paper we prove the uniform convergence of the standard multigrid V-cycle algorithm with Gauss-Seidel relaxation performed only on new nodes and their immediate neighbors for discrete elliptic problems on adaptively refined finite element meshes. The proof depends on sharp estimates on the relationship of local mesh sizes and a new stability estimate for the space decomposition based on Scott-Zhang interpolation operator. Extensive numerical results are reported which confirm the theoretical analysis.