A note on the satisfaction of the boundary conditions for Chebyshev collocation methods in rectangular domains

The way boundary conditions are imposed when applying Chebyshev collocation methods to Poisson and biharmonic-type problems in rectangular domains is investigated. It is shown that careful selection of the number of collocation points leads to a linear system ofn linearly independent equations inn unknowns.     

Conforming spectral methods for Poisson problems in cuboidal domains

A Chebyshev collocation strategy is introduced for the subdivision of cuboids into cuboidal subdomains (elements). These elements are conforming, which means that the approximation to the solution isC ⁰ continuous at all points across their interfaces.     

The method of fundamental solutions for problems in static thermo-elasticity with incomplete boundary data

An inverse problem in static thermo-elasticity is investigated. The aim is to reconstruct the unspecified boundary data, as well as the temperature and displacement inside a body from over-specified boundary data measured on an accessible portion of its boundary. The problem is linear but ill-posed. The uniqueness of the solution is established but the continuous dependence on the input data is violated. In order to reconstruct a stable and accurate solution, the method of fundamental solutions is combined with Tikhonov regularization where the regularization parameter is selected based on the L-curve criterion. Numerical results are presented in both two and three dimensions showing the feasibility and ease of implementation of the proposed technique.     

Conformal Mapping for the Efficient Solution of Poisson Problems with the Kansa-RBF Method

We consider the solution of Poisson Dirichlet problems in simply-connected irregular domains. These domains are conformally mapped onto the unit disk and the resulting Poisson Dirichlet problems are solved efficiently using a Kansa-radial basis function (RBF) method with a matrix decomposition algorithm (MDA). In a similar way, we treat Poisson Dirichlet and Poisson Dirichlet–Neumann problems in doubly-connected domains. These domains are mapped onto annular domains by a conformal mapping and the resulting Poisson Dirichlet and Poisson Dirichlet–Neumann problems are solved efficiently using a Kansa-RBF MDA. Several examples demonstrating the applicability of the proposed technique are presented.     

Efficient MFS Algorithms for Problems in Thermoelasticity

We propose efficient fast Fourier transform (FFT)-based algorithms using the method of fundamental solutions (MFS) for the numerical solution of certain problems in planar thermoelasticity. In particular, we consider problems in domains possessing radial symmetry, namely disks and annuli and it is shown that the MFS matrices arising in such problems possess circulant or block-circulant structures. The solution of the resulting systems is facilitated by appropriately diagonalizing these matrices using FFTs. Numerical experiments demonstrating the applicability of these algorithms are also presented.     

Efficient MFS Algorithms for Inhomogeneous Polyharmonic Problems

In this work we develop an efficient algorithm for the application of the method of fundamental solutions to inhomogeneous polyharmonic problems, that is problems governed by equations of the form Δ ^ℓ u=f, ℓ∈ℕ, in circular geometries. Following the ideas of Alves and Chen (Adv. Comput. Math. 23:125–142, 2005), the right hand side of the equation in question is approximated by a linear combination of fundamental solutions of the Helmholtz equation. A particular solution of the inhomogeneous equation is then easily obtained from this approximation and the resulting homogeneous problem in the method of particular solutions is subsequently solved using the method of fundamental solutions. The fact that both the problem of approximating the right hand side and the homogeneous boundary value problem are performed in a circular geometry, makes it possible to develop efficient matrix decomposition algorithms with fast Fourier transforms for their solution. The efficacy of the method is demonstrated on several test problems.     

Spectral collocation methods for the primary two-point boundary value problem in modelling viscoelastic flows

Expansions in terms of beam functions and Chebyshev polynomials are used to compute solutions to the primary two-point boundary value problem within a spectral collocation formulation. The performance of the methods is analysed in terms of accuracy and robustness relative to the level of non-linearity. Accurate results are obtained with Chebyshev polynomials and the performance of these trial functions is insensitive to the level of non-linearity whereas the behaviour of the beam functions shows great sensitivity to the level of non-linearity. The use of Newton's method to solve the mixed linear-non-linear system for the Chebyshev coefficients is successful for highly non-linear problems without the need for parameter continuation.     

Stress intensity factor computation using the method of fundamental solutions

In this paper, we study the application of the method of fundamental solutions to the computation of stress intensity factors in linear elastic fracture mechanics. The displacements are approximated by linear combinations of the fundamental solutions of the Cauchy–Navier equations of elasticity and the leading terms for the displacement near the crack tip. The applicability of two formulations of the method is demonstrated on two mode I crack problems, where it is shown that accurate approximations for the stress intensity factors can be obtained with relatively few degrees of freedom. Parts of this work were undertaken while the first author was a Visiting Professor in the Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401, U.S.A.     

A numerical conformal mapping method for harmonic mixed boundary value problems

We describe a simple and versatile technique for the numerical solution of harmonic mixed boundary value problems in simply-connected domains. This technique is based on the theory of Riemann-Hilbert problems, and involves only the use of already existing conformal mapping and quadrature routines.