2,487 research outputs found
Space-time adaptive finite elements for nonlocal parabolic variational inequalities
This article considers the error analysis of finite element discretizations
and adaptive mesh refinement procedures for nonlocal dynamic contact and
friction, both in the domain and on the boundary. For a large class of
parabolic variational inequalities associated to the fractional Laplacian we
obtain a priori and a posteriori error estimates and study the resulting
space-time adaptive mesh-refinement procedures. Particular emphasis is placed
on mixed formulations, which include the contact forces as a Lagrange
multiplier. Corresponding results are presented for elliptic problems. Our
numerical experiments for -dimensional model problems confirm the
theoretical results: They indicate the efficiency of the a posteriori error
estimates and illustrate the convergence properties of space-time adaptive, as
well as uniform and graded discretizations.Comment: 47 pages, 20 figure
Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach
The fractional Laplacian is a non-local operator which
depends on the parameter and recovers the usual Laplacian as . A numerical method for the fractional Laplacian is proposed, based on
the singular integral representation for the operator. The method combines
finite difference with numerical quadrature, to obtain a discrete convolution
operator with positive weights. The accuracy of the method is shown to be
. Convergence of the method is proven. The treatment of far
field boundary conditions using an asymptotic approximation to the integral is
used to obtain an accurate method. Numerical experiments on known exact
solutions validate the predicted convergence rates. Computational examples
include exponentially and algebraically decaying solution with varying
regularity. The generalization to nonlinear equations involving the operator is
discussed: the obstacle problem for the fractional Laplacian is computed.Comment: 29 pages, 9 figure
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