Bounds on multivariate polynomials and exponential error estimates for multiquadratic interpolation
Journal of Approximation Theory
SIAM Journal on Numerical Analysis
A Chebyshev collocation method for solving two-phase flow stability problems
Journal of Computational Physics
On the Gibbs Phenomenon and Its Resolution
SIAM Review
The immersed interface method using a finite element formulation
Applied Numerical Mathematics
On the computation of high order pseudospectral derivatives
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
Overlapping domain decomposition method by radial basis functions
Applied Numerical Mathematics
Least-squares spectral collocation for discontinuous and singular perturbation problems
Journal of Computational and Applied Mathematics
Pseudospectral Least-Squares Method for the Second-Order Elliptic Boundary Value Problem
SIAM Journal on Numerical Analysis
Radial Basis Functions
Least-Squares Spectral Collocation for the Navier–Stokes Equations
Journal of Scientific Computing
Numerical approximations of singular source terms in differential equations
Journal of Computational Physics
Polynomials and Potential Theory for Gaussian Radial Basis Function Interpolation
SIAM Journal on Numerical Analysis
Discretization of Dirac delta functions in level set methods
Journal of Computational Physics
A note on the Gibbs phenomenon with multiquadric radial basis functions
Applied Numerical Mathematics
Computers & Mathematics with Applications
The Runge phenomenon and spatially variable shape parameters in RBF interpolation
Computers & Mathematics with Applications
Highly accurate finite element method for one-dimensional elliptic interface problems
Applied Numerical Mathematics
Journal of Computational Physics
Meshfree Approximation Methods with MATLAB
Meshfree Approximation Methods with MATLAB
Journal of Scientific Computing
Adaptive radial basis function methods for time dependent partial differential equations
Applied Numerical Mathematics
Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous Problems
Journal of Scientific Computing
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Differential equations with singular sources or discontinuous coefficients yield non-smooth or even discontinuous solutions. This problem is known as the interface problem. High-order numerical solutions suffer from the Gibbs phenomenon in that the accuracy deteriorates if the discontinuity is not properly treated. In this work, we use the spectral and radial basis function methods and present a least squares collocation method to solve the interface problem for one-dimensional elliptic equations. The domain is decomposed into multiple sub-domains; in each sub-domain, the collocation solution is sought. The solution should satisfy more conditions than the given conditions associated with the differential equations, which makes the problem over-determined. To solve the over-determined system, the least squares method is adopted. For the spectral method, the weighted norm method with different scaling factors and the mixed formulation are used. For the radial basis function method, the weighted shape parameter method is presented. Numerical results show that the least squares collocation method provides an accurate solution with high efficacy and that better accuracy is obtained with the spectral method.