52 research outputs found

    Adaptive meshless centres and RBF stencils for Poisson equation

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    We consider adaptive meshless discretisation of the Dirichlet problem for Poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions. New meshless stencil selection and adaptive refinement algorithms are proposed in 2D. Numerical experiments show that the accuracy of the solution is comparable with, and often better than that achieved by the mesh-based adaptive finite element method

    On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation

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    We investigate the influence of the shape parameter in the meshless Gaussian RBF finite difference method with irregular centres on the quality of the approximation of the Dirichlet problem for the Poisson equation with smooth solution. Numerical experiments show that the optimal shape parameter strongly depends on the problem, but insignificantly on the density of the centres. Therefore, we suggest a multilevel algorithm that effectively finds near-optimal shape parameter, which helps to significantly reduce the error. Comparison to the finite element method and to the generalised finite differences obtained in the flat limits of the Gaussian RBF is provided

    Adaptive meshless refinement schemes for RBF-PUM collocation

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    In this paper we present an adaptive discretization technique for solving elliptic partial differential equations via a collocation radial basis function partition of unity method. In particular, we propose a new adaptive scheme based on the construction of an error indicator and a refinement algorithm, which used together turn out to be ad-hoc strategies within this framework. The performance of the adaptive meshless refinement scheme is assessed by numerical tests

    Adaptive radial basis function generated finite-difference (RBF-FD) on non-uniform nodes using pp-refinement

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    Radial basis functions-generated finite difference methods (RBF-FDs) have been gaining popularity recently. In particular, the RBF-FD based on polyharmonic splines (PHS) augmented with multivariate polynomials (PHS+poly) has been found significantly effective. For the approximation order of RBF-FDs' weights on scattered nodes, one can already find mathematical theories in the literature. Many practical problems in numerical analysis, however, do not require a uniform node-distribution. Instead, they would be better suited if specific areas of the domain, where complicated physics needed to be resolved, had a relatively higher node-density compared to the rest of the domain. In this work, we proposed a practical adaptive RBF-FD with a user-defined order of convergence with respect to the total number of (possibly scattered and non-uniform) data points NN. Our algorithm outputs a sparse differentiation matrix with the desired approximation order. Numerical examples are provided to show that the proposed adaptive RBF-FD method yields the expected NN-convergence even for highly non-uniform node-distributions. The proposed method also reduces the number of non-zero elements in the linear system without sacrificing accuracy.Comment: An updated version with seismic modeling will be included in version

    Solución bidimensional sin malla de la ecuación no lineal de convección-difusión-reacción mediante el método de Interpolación Local Hermítica

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    A meshless numerical scheme is developed for solving a generic version of the non-linear convection-diffusion-reaction equation in two-dimensional domains. The Local Hermitian Interpolation (LHI) method is employed for the spatial discretization and several strategies are implemented for the solution of the resulting non-linear equation system, among them the Picard iteration, the Newton Raphson method and a truncated version of the Homotopy Analysis Method (HAM). The LHI method is a local collocation strategy in which Radial Basis Functions (RBFs) are employed to build the interpolation function. Unlike the original Kansa’s Method, the LHI is applied locally and the boundary and governing equation differential operators are used to obtain the interpolation function, giving a symmetric and non-singular collocation matrix. Analytical and Numerical Jacobian matrices are tested for the Newton-Raphson method and the derivatives of the governing equation with respect to the homotopy parameter are obtained analytically. The numerical scheme is verified by comparing the obtained results to the one-dimensional Burgers’ and two-dimensional Richards’ analytical solutions. The same results are obtained for all the non-linear solvers tested, but better convergence rates are attained with the Newton Raphson method in a double iteration scheme.Se desarrolla un esquema numérico sin malla para resolver una versión genérica de la ecuación no lineal de convección-difusión-reacción en dominios bidimensionales. El método de Interpolación Hermitiana Local (LHI) se emplea para la discretización espacial y se implementan varias estrategias para la solución del sistema de ecuaciones no lineal resultante, entre ellas la iteración Picard, el método Newton Raphson y una versión truncada del Método de Análisis de Homotopía. (JAMÓN). El método LHI es una estrategia de colocación local en la que se utilizan funciones de base radial (RBF) para construir la función de interpolación. A diferencia del método original de Kansa, el LHI se aplica localmente y los operadores diferenciales de ecuación límite y gobernante se utilizan para obtener la función de interpolación, dando una matriz de colocación simétrica y no singular. Las matrices analíticas y numéricas jacobianas se prueban para el método de Newton-Raphson y las derivadas de la ecuación de gobierno con respecto al parámetro de homotopía se obtienen analíticamente. El esquema numérico se verifica comparando los resultados obtenidos con las soluciones analíticas unidimensionales de Burgers y Richards bidimensionales. Se obtienen los mismos resultados para todos los solucionadores no lineales probados, pero se obtienen mejores tasas de convergencia con el método Newton Raphson en un esquema de doble iteración

    RBF-Based Partition of Unity Methods for Elliptic PDEs: Adaptivity and Stability Issues Via Variably Scaled Kernels

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    We investigate adaptivity issues for the approximation of Poisson equations via radial basis function-based partition of unity collocation. The adaptive residual subsampling approach is performed with quasi-uniform node sequences leading to a flexible tool which however might suffer from numerical instability due to ill-conditioning of the collocation matrices. We thus develop a hybrid method which makes use of the so-called variably scaled kernels. The proposed algorithm numerically ensures the convergence of the adaptive procedure

    Optimal variable shape parameter for multiquadric based RBF-FD method

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    In this follow up paper to our previous study in Bayona et al. (2011) [2], we present a new technique to compute the solution of PDEs with the multiquadric based RBF finite difference method (RBF-FD) using an optimal node dependent variable value of the shape parameter. This optimal value is chosen so that, to leading order, the local approximation error of the RBF-FD formulas is zero. In our previous paper (Bayona et al., 2011) [2] we considered the case of an optimal (constant) value of the shape parameter for all the nodes. Our new results show that, if one allows the shape parameter to be different at each grid point of the domain, one may obtain very significant accuracy improvements with a simple and inexpensive numerical technique. We analyze the same examples studied in Bayona et al. (2011) [2], both with structured and unstructured grids, and compare our new results with those obtained previously. We also find that, if there are a significant number of nodes for which no optimal value of the shape parameter exists, then the improvement in accuracy deteriorates significantly. In those cases, we use generalized multiquadrics as RBFs and choose the exponent of the multiquadric at each node to assure the existence of an optimal variable shape parameter.This work has been supported by Spanish MICINN Grants FIS2010-18473, CSD2010-00011 and by Madrid Autonomous Region grant S2009-1597. M.K. acknowledges Fundacion Caja Madrid for its financial support
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