14 research outputs found
A high-order approximation method for semilinear parabolic equations on spheres
We describe a novel discretisation method for numerically solving (systems of) semilinear parabolic equations on Euclidean spheres. The new approximation method is based upon a discretisation in space using spherical basis functions and can be of arbitrary order. This, together with the fact that the solutions of semilinear parabolic problems are known to be infinitely smooth, at least locally in time, allows us to prove stability and convergence of the discretisation in a straight-forward way
A kernel-based meshless conservative Galerkin method for solving Hamiltonian wave equations
We propose a meshless conservative Galerkin method for solving Hamiltonian
wave equations. We first discretize the equation in space using radial basis
functions in a Galerkin-type formulation. Differ from the traditional RBF
Galerkin method that directly uses nonlinear functions in its weak form, our
method employs appropriate projection operators in the construction of the
Galerkin equation, which will be shown to conserve global energies. Moreover,
we provide a complete error analysis to the proposed discretization. We further
derive the fully discretized solution by a second order average vector field
scheme. We prove that the fully discretized solution preserved the discretized
energy exactly. Finally, we provide some numerical examples to demonstrate the
accuracy and the energy conservation
A kernel-based least-squares collocation method for surface diffusion
There are plenty of applications and analysis for time-independent elliptic
partial differential equations in the literature hinting at the benefits of
overtesting by using more collocation conditions than the number of basis
functions. Overtesting not only reduces the problem size, but is also known to
be necessary for stability and convergence of widely used unsymmetric
Kansa-type strong-form collocation methods. We consider kernel-based meshfree
methods, which is a method of lines with collocation and overtesting spatially,
for solving parabolic partial differential equations on surfaces without
parametrization. In this paper, we extend the time-independent convergence
theories for overtesting techniques to the parabolic equations on smooth and
closed surfaces.Comment: 4 figures, 21 page
A Nash-Hormander iteration and boundary elements for the Molodensky problem
We investigate the numerical approximation of the nonlinear Molodensky
problem, which reconstructs the surface of the earth from the gravitational
potential and the gravity vector. The method, based on a smoothed
Nash-Hormander iteration, solves a sequence of exterior oblique Robin problems
and uses a regularization based on a higher-order heat equation to overcome the
loss of derivatives in the surface update. In particular, we obtain a
quantitative a priori estimate for the error after m steps, justify the use of
smoothing operators based on the heat equation, and comment on the accurate
evaluation of the Hessian of the gravitational potential on the surface, using
a representation in terms of a hypersingular integral. A boundary element
method is used to solve the exterior problem. Numerical results compare the
error between the approximation and the exact solution in a model problem.Comment: 32 pages, 14 figures, to appear in Numerische Mathemati
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Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018
This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions