Improved Kansa RBF method for the solution of nonlinear boundary value problems

Jankowska, Małgorzata A.; Karageorghis, Andréas; Chen, Ching-Shyang

doi:10.1016/j.enganabound.2017.11.012

Cited by 53 publications

(22 citation statements)

References 19 publications

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“…We assume to satisfy the assumptions of the above theorem, and we shall apply it to quasilinear elliptic problems with Dirichlet bundary conditions in Section 7. In this case, the PDE literature mainly works on Hölder spaces, using  = 2, and  = 0, (Ω) × 0, ( Ω), see (13) below, and this causes some problems when we follow [22] and [5]. We shall deal with this issue in Section 7, but for now we go on with the framework we need.…”

Section: Well-posed Nonlinear Problemsmentioning

confidence: 99%

A nonlinear discretization theory for meshfree collocation methods applied to quasilinear elliptic equations

Böhmer

Schaback²

2020

Z Angew Math Mech

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This paper generalizes earlier results concerning meshfree collocation methods for semilinear elliptic second order problems to the quasilinear case. These results require a strong form of well-posedness and a stability analysis in the ∞ norm for the meshfree discretization. Since quasilinear problems are usually treated in Hölder norms, both aspects have to be transformed to the Hölder situation. This finally leads to error bounds and convergence rates, the latter being dependent on the smoothness of the true solution. The paper focuses on Dirichlet conditions and postpones other boundary conditions to a forthcoming paper. A numerical example is provided.

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Section: Well-posed Nonlinear Problemsmentioning

confidence: 99%

A nonlinear discretization theory for meshfree collocation methods applied to quasilinear elliptic equations

Böhmer

Schaback²

2020

Z Angew Math Mech

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“…In this way, constraints on the values of the coefficients and in particular on the values of the shape parameter may be imposed. (13) were also calculated. In Example 1, both MQ (3) and IMQ (4) were used while in all the other numerical examples only MQ were used.…”

Section: Augmented Polynomial Basismentioning

confidence: 99%

“…was taken. In addition to the errors (12) and (13), c min and c max were calculated, which are the minimum and maximum values, respectively, of the entries of the final vector c.…”

Section: Examplementioning

confidence: 99%

“…In this work the problem of approximating functions in two-dimensional domains using an RBF collocation method is considered and an attempt to address the issue of estimating an appropriate value of the RBF shape parameter is made. The basic idea was introduced in [12,13] where the Kansa-RBF method [14] was applied to two-dimensional second and fourth order nonlinear boundary value problems. The shape parameter determination issue is still present in such applications and to overcome it, in [12,13] it (the shape parameter) was taken to be one of the unknowns to be determined as part of the solution.…”

Section: Introductionmentioning

confidence: 99%

“…The basic idea was introduced in [12,13] where the Kansa-RBF method [14] was applied to two-dimensional second and fourth order nonlinear boundary value problems. The shape parameter determination issue is still present in such applications and to overcome it, in [12,13] it (the shape parameter) was taken to be one of the unknowns to be determined as part of the solution. Since the problems in question were nonlinear anyway, this inclusion did not affect the discretization of the problem much as this added nonlinearity was absorbed, in a natural way, by the resulting nonlinear system.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Shape parameter estimation in RBF function approximation

Karageorghis¹,

Tryfonos²

2019

Int. J. CMEM

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The radial basis function (RBF) collocation method is applied for the approximation of functions in two variables. When the RBFs employed include a shape parameter, the determination of an appropriate value for it is a major issue. In this work, this is addressed by including the value of the shape parameter in the unknowns along with the coefficients of the RBFs in the approximation. The variable shape parameter case when a different shape parameter is associated with each RBF in the approximation is also considered. Both approaches yield nonlinear systems of equations, which are solved by a standard non-linear solver. The results of several numerical experiments are presented.

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The method of fundamental solutions for the Oseen steady‐state viscous flow past obstacles of known or unknown shapes

Karageorghis

Lesnic

2019

Numerical Methods Partial

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In this paper, the steady-state Oseen viscous flow equations past a known or unknown obstacle are solved numerically using the method of fundamental solutions (MFS), which is free of meshes, singularities, and numerical integrations. The direct problem is linear and well-posed, whereas the inverse problem is nonlinear and ill-posed. For the direct problem, the MFS computations of the fluid flow characteristics (velocity, pressure, drag, and lift coefficients) are in very good agreement with the previously published results obtained using other methods for the Oseen flow past circular and elliptic cylinders, as well as past two circular cylinders. In the inverse obstacle problem the boundary data and the internal measurement of the fluid velocity are minimized using the MATLAB © optimization toolbox lsqnonlin routine. Regularization was found necessary in the case the measured data are contaminated with noise. Numerical results show accurate and stable reconstructions of various star-shaped obstacles of circular, bean, or peanut cross-section.

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Improved Kansa RBF method for the solution of nonlinear boundary value problems

Cited by 53 publications

References 19 publications

A nonlinear discretization theory for meshfree collocation methods applied to quasilinear elliptic equations

A nonlinear discretization theory for meshfree collocation methods applied to quasilinear elliptic equations

Shape parameter estimation in RBF function approximation

The method of fundamental solutions for the Oseen steady‐state viscous flow past obstacles of known or unknown shapes

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