1,978 research outputs found
On a nonlocal degenerate parabolic problem
Conditions for the existence and uniqueness of weak solutions for a class of
nonlinear nonlocal degenerate parabolic equations are established. The
asymptotic behaviour of the solutions as time tends to infinity are also
studied. In particular, the finite time extinction and polynomial decay
properties are proved
Convergence of the Crank-Nicolson-Galerkin finite element method for a class of nonlocal parabolic systems with moving boundaries
The aim of this paper is to establish the convergence and error bounds to the
fully discrete solution for a class of nonlinear systems of reaction-diffusion
nonlocal type with moving boundaries, using a linearized
Crank-Nicolson-Galerkin finite element method with polynomial approximations of
any degree. A coordinate transformation which fixes the boundaries is used.
Some numerical tests to compare our Matlab code with some existing moving
finite elements methods are investigated
Homogenization of Parabolic Equations with a Continuum of Space and Time Scales
This paper addresses the issue of the homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available. Namely, we consider divergence form linear parabolic operators in with -coefficients. It appears that the inverse operator maps the unit ball of into a space of functions which at small (time and space) scales are close in norm to a functional space of dimension . It follows that once one has solved these equations at least times it is possible to homogenize them both in space and in time, reducing the number of operation counts necessary to obtain further solutions. In practice we show under a Cordes-type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in (instead of with Euclidean coordinates). If the medium is time-independent, then it is sufficient to solve times the associated elliptic equation in order to homogenize the parabolic equation
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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