1,818 research outputs found
An L1 Penalty Method for General Obstacle Problems
We construct an efficient numerical scheme for solving obstacle problems in
divergence form. The numerical method is based on a reformulation of the
obstacle in terms of an L1-like penalty on the variational problem. The
reformulation is an exact regularizer in the sense that for large (but finite)
penalty parameter, we recover the exact solution. Our formulation is applied to
classical elliptic obstacle problems as well as some related free boundary
problems, for example the two-phase membrane problem and the Hele-Shaw model.
One advantage of the proposed method is that the free boundary inherent in the
obstacle problem arises naturally in our energy minimization without any need
for problem specific or complicated discretization. In addition, our scheme
also works for nonlinear variational inequalities arising from convex
minimization problems.Comment: 20 pages, 18 figure
A Modular Regularized Variational Multiscale Proper Orthogonal Decomposition for Incompressible Flows
In this paper, we propose, analyze and test a post-processing implementation
of a projection-based variational multiscale (VMS) method with proper
orthogonal decomposition (POD) for the incompressible Navier-Stokes equations.
The projection-based VMS stabilization is added as a separate post-processing
step to the standard POD approximation, and since the stabilization step is
completely decoupled, the method can easily be incorporated into existing
codes, and stabilization parameters can be tuned independent from the time
evolution step. We present a theoretical analysis of the method, and give
results for several numerical tests on benchmark problems which both illustrate
the theory and show the proposed method's effectiveness
A Multigrid Optimization Algorithm for the Numerical Solution of Quasilinear Variational Inequalities Involving the -Laplacian
In this paper we propose a multigrid optimization algorithm (MG/OPT) for the
numerical solution of a class of quasilinear variational inequalities of the
second kind. This approach is enabled by the fact that the solution of the
variational inequality is given by the minimizer of a nonsmooth energy
functional, involving the -Laplace operator. We propose a Huber
regularization of the functional and a finite element discretization for the
problem. Further, we analyze the regularity of the discretized energy
functional, and we are able to prove that its Jacobian is slantly
differentiable. This regularity property is useful to analyze the convergence
of the MG/OPT algorithm. In fact, we demostrate that the algorithm is globally
convergent by using a mean value theorem for semismooth functions. Finally, we
apply the MG/OPT algorithm to the numerical simulation of the viscoplastic flow
of Bingham, Casson and Herschel-Bulkley fluids in a pipe. Several experiments
are carried out to show the efficiency of the proposed algorithm when solving
this kind of fluid mechanics problems
Heuristic parameter-choice rules for convex variational regularization based on error estimates
In this paper, we are interested in heuristic parameter choice rules for
general convex variational regularization which are based on error estimates.
Two such rules are derived and generalize those from quadratic regularization,
namely the Hanke-Raus rule and quasi-optimality criterion. A posteriori error
estimates are shown for the Hanke-Raus rule, and convergence for both rules is
also discussed. Numerical results for both rules are presented to illustrate
their applicability
An Energy-Minimization Finite-Element Approach for the Frank-Oseen Model of Nematic Liquid Crystals: Continuum and Discrete Analysis
This paper outlines an energy-minimization finite-element approach to the
computational modeling of equilibrium configurations for nematic liquid
crystals under free elastic effects. The method targets minimization of the
system free energy based on the Frank-Oseen free-energy model. Solutions to the
intermediate discretized free elastic linearizations are shown to exist
generally and are unique under certain assumptions. This requires proving
continuity, coercivity, and weak coercivity for the accompanying appropriate
bilinear forms within a mixed finite-element framework. Error analysis
demonstrates that the method constitutes a convergent scheme. Numerical
experiments are performed for problems with a range of physical parameters as
well as simple and patterned boundary conditions. The resulting algorithm
accurately handles heterogeneous constant coefficients and effectively resolves
configurations resulting from complicated boundary conditions relevant in
ongoing research.Comment: 31 pages, 3 figures, 3 table
A Duality Approach to Error Estimation for Variational Inequalities
Motivated by problems in contact mechanics, we propose a duality approach for
computing approximations and associated a posteriori error bounds to solutions
of variational inequalities of the first kind. The proposed approach improves
upon existing methods introduced in the context of the reduced basis method in
two ways. First, it provides sharp a posteriori error bounds which mimic the
rate of convergence of the RB approximation. Second, it enables a full
offline-online computational decomposition in which the online cost is
completely independent of the dimension of the original (high-dimensional)
problem. Numerical results comparing the performance of the proposed and
existing approaches illustrate the superiority of the duality approach in cases
where the dimension of the full problem is high.Comment: 21 pages, 8 figure
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