43 research outputs found
Adaptive monotone multigrid methods for nonlinear variational problems
A wide range of problems occurring in engineering and industry is characterized by the presence of a free (i.e. a priori unknown) boundary where the underlying physical situation is changing in a discontinuous way. Mathematically, such phenomena can be often reformulated as variational inequalities or related non–smooth minimization problems.
In these research notes, we will describe a new and promising way of constructing fast solvers for the corresponding discretized problems providing globally convergent iterative schemes with (asymptotic) multigrid
convergence speed. The presentation covers physical modelling, existence and uniqueness results, finite element approximation and adaptive mesh–refinement based on a posteriori error estimation. The numerical properties
of the resulting adaptive multilevel algorithm are illustrated by typical applications, such as semiconductor device simulation or continuous casting
Numerical Computations with H(div)-Finite Elements for the Brinkman Problem
The H(div)-conforming approach for the Brinkman equation is studied
numerically, verifying the theoretical a priori and a posteriori analysis in
previous work of the authors. Furthermore, the results are extended to cover a
non-constant permeability. A hybridization technique for the problem is
presented, complete with a convergence analysis and numerical verification.
Finally, the numerical convergence studies are complemented with numerical
examples of applications to domain decomposition and adaptive mesh refinement.Comment: Minor clarifications, added references. Reordering of some figures.
To appear in Computational Geosciences, final article available at
http://www.springerlink.co
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On the Babuška-Osborn approach to finite element analysis: L2 estimates for unstructured meshes
The standard approach to L2 bounds uses theH1 bound in combination to a duality argument, known as Nitsche’s trick, to recover the optimal a priori order of the method. Although this approach makes perfect sense for quasi-uniform meshes, it does not provide the expected information for unstructured meshes since the final estimate involves the maximum mesh size. Babuška and Osborn, [1], addressed this issue for a one dimensional problem by introducing a technique based on mesh-dependent norms. The key idea was to see the bilinear form posed on two different spaces; equipped with the mesh dependent analogs of L2 and H2 and to show that the finite element space is inf-sup stable with respect to these norms. Although this approach is readily extendable to multidimensional setting, the proof of the inf-sup stability with respect to mesh dependent norms is known only in very limited cases. We establish the validity of the inf-sup condition for standard conforming finite element spaces of any polynomial degree under certain restrictions on the mesh variation which however permit unstructured non quasiuniform meshes. As a consequence we derive L2 estimates for the finite element approximation via quasioptimal bounds and examine related stability properties of the elliptic projection
Mathematical sciences : solution procedures for three-dimensional crack problems in elasticity : boundary integral equations and boundary elements
Issued as Final report, Project no. G-37-63
Fast computations with the harmonic Poincaré-Steklov operators on nested refined meshes
In this paper we develop asymptotically optimal algorithms for fast computations with the discrete harmonic Poincar'e-Steklov operators in presence of nested mesh refinement. For both interior and exterior problems the matrix-vector multiplication for the finite element approximations to the Poincar'e-Steklov operators is shown to have a complexity of the order O(Nreflog3N) where Nref is the number of degrees of freedom on the polygonal boundary under consideration and N = 2-p0 · Nref, p0 ≥ 1, is the dimension of a finest quasi-uniform level. The corresponding memory needs are estimated by O(Nreflog2N). The approach is based on the multilevel interface solver (as in the case of quasi-uniform meshes, see [20]) applied to the Schur complement reduction onto the nested refined interface associated with nonmatching decomposition of a polygon by rectangular substructures