Computer Methods in Applied Mechanics and Engineering
An analytical and numerical study of the two-dimensional Bratu equation
Journal of Scientific Computing
A group-theoretic approach to computational bifurcation problems with symmetry
Computer Methods in Applied Mechanics and Engineering
Boundary value problems with symmetry and their approximation by finite elements
SIAM Journal on Applied Mathematics
Eigenvalues for equivariant matrices
Journal of Computational and Applied Mathematics - Special issue on computational and mathematical methods in science and engineering (CMMSE-2004)
Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform
Journal of Computational and Applied Mathematics
Meshfree Approximation Methods with MATLAB
Meshfree Approximation Methods with MATLAB
Error saturation in Gaussian radial basis functions on a finite interval
Journal of Computational and Applied Mathematics
Journal of Computational Physics
Hi-index | 31.45 |
A domain is invariant under the eight-element D"4 symmetry group if it is unchanged by reflection with respect to the x and y axes and also the diagonal line x=y. Previous treatments of group theory for spectral methods have generally demanded a semester@?s worth of group theory. We show this is unnecessary by providing explicit recipes for creating grids, etc. We show how to decompose an arbitrary function into six symmetry-invariant components, and thereby split the interpolation problem into six independent subproblems. We also show how to make symmetry-invariant basis functions from products of Chebyshev polynomials, from Zernike polynomials and from radial basis functions (RBFs) of any species. These recipes are completely general, and apply to any domain that is invariant under the dihedral group D"4. These concepts are illustrated by RBF pseudospectral solutions of the Poisson equation in a domain bounded by a squircle, the square-with-rounded corners defined by x^2^@n+y^2^@n-1=0 where here @n=2. We also apply Chebyshev polynomials to compute eigenmodes of the Helmholtz equation on the square and show each mode belongs to one and only one of the six D"4 classes.