An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometries

Tominec, Igor; Breznik, Eva

doi:10.1016/j.jcp.2021.110283

Cited by 22 publications

(12 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A related work is [25] where we introduced an oversampled version of the RBF-FD method for an elliptic model problem, and made a theoretical investigation of its stability properties, where we showed that the RBF-FD method generates a discontinuous trial space and that oversampling of a stationary problem approximates a variational formulation using a least-squares projection. The oversampled RBF-FD method was then also extended to an unfitted setting in [24], which was later used for the simulations of a thoracic diaphragm in [27]. In [26] we introduced the oversampled RBF-FD method to solve nonlinear conservation laws with discontinuous solutions, and found that oversampling approximates a strong variational formulation using a Galerkin projection, but the method is still unstable in time.…”

mentioning

confidence: 99%

Stability estimates for radial basis function methods applied to time-dependent hyperbolic PDEs

Tominec¹,

Nazarov²,

Larsson³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We derive stability estimates for three commonly used radial basis function (RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and Kansa's (global) RBF method. We give the estimates in the discrete 2 -norm intrinsic to each of the three methods. The results show that Kansa's method and RBF-PUM can be 2 -stable in time under a sufficiently large oversampling of the discretized system of equations. On the other hand, the RBF-FD method is not 2 -stable by construction, no matter how large the oversampling is. We show that this is due to the jumps (discontinuities) in the RBF-FD cardinal basis functions. We also provide a stabilization of the RBF-FD method that penalizes the spurious jumps. Numerical experiments show an agreement with our theoretical observations.

show abstract

mentioning

confidence: 99%

Stability estimates for radial basis function methods applied to time-dependent hyperbolic PDEs

Tominec¹,

Nazarov²,

Larsson³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In Figure 2 we report the condition number κ(M) of the interpolation matrix and the constant Λ I , defined in equation (24), against the angle α in the case of the reference stencil of Figure 1. Whenever one of these quantities grows unbounded, we have a very ill-conditioned system in equation (17), which translates into large errors and most likely stability issues.…”

Section: Preliminary Considerationsmentioning

confidence: 99%

“…In [22,23,24] the stabilization is achieved through different least squares procedures. The one in [24] seems especially capable of providing both stable and accurate results by using two different node sets.…”

Section: Introductionmentioning

confidence: 99%

Accurate Stabilization Techniques for Rbf-Fd Meshless Discretizations with Neumann Boundary Conditions

Zamolo

Miotti

2022

SSRN Journal

View full text Add to dashboard Cite

“…A mathematical formulation for computing these weights is given in [33,8]. In this paper we use these approaches to generate a vector of weights for every y ∈ Y , such that the weights are exact for a cubic polyharmonic spline basis and a monomial basis of degree p, which is a concept introduced in [34] and studied in [35,36,37,38].…”

Section: Evaluation and Differentiation Matricesmentioning

confidence: 99%

“…In Figure 16 and Figure 17, we display the convergence under node refinement, in 2-norm and 1norm respectively, where the norms are defined in (33). The optimal convergence rates are expected to be 0.5 and 1.…”

Section: Numerical Study Ii: Burger's Equationmentioning

confidence: 99%

Residual viscosity stabilized RBF-FD methods for solving nonlinear conservation laws

Tominec¹,

Nazarov²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We formulate an oversampled radial basis function generated finite difference (RBF-FD) method to solve time-dependent nonlinear conservation laws. The analytic solutions of these problems are known to be discontinuous, which leads to occurrence of non-physical oscillations (Gibbs phenomenon) that pollute the numerical solutions and can make them unstable. We address these difficulties using a residual based artificial viscosity stabilization, where the residual of the conservation law indicates the approximate location of the shocks. The location is then used to locally apply an upwind viscosity term, which stabilizes the Gibbs phenomenon and does not smear the solution away from the shocks. The proposed method is numerically tested and proves to be robust and accurate when solving scalar conservation laws and systems of conservation laws, such as compressible Euler equations.

show abstract

An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometries

Cited by 22 publications

References 28 publications

Stability estimates for radial basis function methods applied to time-dependent hyperbolic PDEs

Stability estimates for radial basis function methods applied to time-dependent hyperbolic PDEs

Accurate Stabilization Techniques for Rbf-Fd Meshless Discretizations with Neumann Boundary Conditions

Residual viscosity stabilized RBF-FD methods for solving nonlinear conservation laws

Contact Info

Product

Resources

About