1,317 research outputs found
A bipolynomial fractional Dirichlet-Laplace problem
In the paper, we derive an existence result for a nonlinear nonautonomous
partial elliptic system on an open bounded domain with Dirichlet boundary
conditions, containg fractional powers of the weak Dirichlet-Laplace operator
that are meant in the Stone-von Neumann operator calculus sense. We apply a
variational method which gives strong solutions of the problem under
consideration
Reliable and efficient a posteriori error estimates for finite element approximations of the parabolic p-Laplacian
We generalize the a posteriori techniques for the linear heat equation in [Ver03] to the case of the nonlinear parabolic p-Laplace problem thereby proving reliable and efficient a posteriori error estimates for a fully discrete implicite Euler Galerkin finite element scheme. The error is analyzed using the so-called quasi-norm and a related dual error expression. This leads to equivalence of the error and the residual, which is the key property for proving the error bounds
Vibrational modes of circular free plates under tension
The vibrational frequencies of a plate under tension are given by the
eigenvalues of the equation . This
paper determines the eigenfunctions and eigenvalues of this bi-Laplace problem
on the ball under natural (free) boundary conditions. In particular, the
fundamental modes --- the eigenfunctions of the lowest nonzero eigenvalue ---
are identified and found to have simple angular dependence.Comment: 17 pages. To be submitted for publication shortly
A higher-order singularity subtraction technique for the discretization of singular integral operators on curved surfaces
This note is about promoting singularity subtraction as a helpful tool in the
discretization of singular integral operators on curved surfaces. Singular and
nearly singular kernels are expanded in series whose terms are integrated on
parametrically rectangular regions using high-order product integration,
thereby reducing the need for spatial adaptivity and precomputed weights. A
simple scheme is presented and an application to the interior Dirichlet Laplace
problem on some tori gives around ten digit accurate results using only two
expansion terms and a modest programming- and computational effort.Comment: 7 pages, 2 figure
Boundedness of the gradient of a solution to the Neumann-Laplace problem in a convex domain
It is shown that solutions of the Neumann problem for the Poisson equation in
an arbitrary convex -dimensional domain are uniformly Lipschitz.
Applications of this result to some aspects of regularity of solutions to the
Neumann problem on convex polyhedra are given
Spectral methods for exterior elliptic problems
Spectral approximations for exterior elliptic problems in two dimensions are discussed. As in the conventional finite difference or finite element methods, the accuracy of the numerical solutions is limited by the order of the numerical farfield conditions. A spectral boundary treatment is introduced at infinity which is compatible with the infinite order interior spectral scheme. Computational results are presented to demonstrate the spectral accuracy attainable. Although a simple Laplace problem is examined, the analysis covers more complex and general cases
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