Convergence of Adaptive Finite Element Methods

• Autores: Morin, Pedro; Nochetto, Ricardo Horacio; Siebert, Kunibert G.
• Año de publicación: 2002
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: A POSTERIORI ERROR ESTIMATORS; ADAPTIVE MESH REFINEMENT; CONVERGENCE; DATA OSCILLATION; STOKES PROBLEM; UZAWA ITERATIONS
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

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; Argentina
Fil: Nochetto, Ricardo Horacio. University of Maryland; Estados Unidos
Fil: Siebert, Kunibert G.. Universität Heidelberg;