Convergence of Adaptive Finite Element Methods

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.Fil: Morin, Pedro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Nochetto, Ricardo Horacio. University of Maryland; Estados UnidosFil: Siebert, Kunibert G.. Universität Heidelberg

Experimental and numerical investigation of edge tones

We study both, by experimental and numerical means the fluid dynamical phenomenon of so-called edge tones. Of particular interest is the clarification of certain scaling laws relating the frequency ƒ to geometrical quantities, namely 푑, the height of the jet, 푤, the stand-off distance and the velocity of the jet. We conclude that the Strouhal number S푑 is given by S푑 = C · (푑/푤)푛 with 푛 ≈ 1 in our case. Moreover, the constant C of the experiment agrees within 10-15% with the result of the simulation. As for the frequency dependence on the geometry and on the jet velocity there is a very good agreement of experimental and numerical results

A Basic Convergence Result for Conforming Adaptive Finite Elements

We consider the approximate solution with adaptive finite elements of a class of linear boundary value problems, which includes problems of 'saddle point' type. For the adaptive algorithm we suppose the following framework: refinement relies on unique quasi-regular element subdivisions and generates locally quasi-uniform grids, the finite element spaces are conforming, nested, and satisfy the inf-sup conditions, the error estimator is reliable as well as locally and discretely efficient, and marked elements are subdivided at least once. Under these assumptions, we give a sufficient and essentially necessary condition on marking for the convergence of the finite element solutions to the exact one. This condition is not only satisfied by Dörfler's strategy, but also by the maximum strategy and the equidistribution strategy

Design of Finite Element Tools for Coupled Surface and Volume Meshes

Many problems with underlying variational structure involve a coupling of volume with surface effects. A straight-forward approach in a finite element discretization is to make use of the surface triangulation that is naturally induced by the volume triangulation. In an adaptive method one wants to facilitate "matching" local mesh modifications, i. e., local refinement and/or coarsening, of volume and surface mesh with standard tools such that the surface grid is always induced by the volume grid. We describe the concepts behind this approach for bisectional refinement and describe new tools incorporated in the finite element toolbox ALBERTA. We also present several important applications of the mesh coupling

Convergence of Finite Elements Adapted for Weak Norms

We consider finite elements that are adapted to a (semi)norm that is weaker than the one of the trial space. We establish convergence of the finite element solutions to the exact one under the following conditions: refinement relies on unique quasi-regular element subdivisions and generates locally quasi-uniform grids; the finite element spaces are conforming, nested, and satisfy the inf-sup condition; the error estimator is reliable and appropriately locally efficient; the indicator of a non-marked element is bounded by the estimator contribution associated with the marked elements, and each marked element is subdivided at least once. This abstract convergence result is illustrated by two examples

Multiresolution visualization of higher order adaptive finite element simulations

Multiresolution visualization of higher order adaptive finite element simulations / K. G. Siebert ... - In: Computing. 70. 2003. S. 181-20

Local problems on stars

Local problems on stars : a posteriori error estimators, convergence, and performance / P. Morin, R. H. Nochetto, K. G. Siebert. - In: Mathematics of computation. 72 . 2003. S. 1067-109

Design of adaptive finite element software

Design of adaptive finite element software : the finite element toolbox ALBERTA / Alfred Schmidt ; Kunibert G. Siebert. - Berlin u.a. : Springer, 2005. - XII, 315 S. + 1 CD-ROM. - (Lecture notes in computational science and engineering ; 42