A stable meshfree PDE solver for source-type flows in porous media

Abstract

An elliptic partial differential equation with a singular forcing term, describing a steady state flow determined by a pulse-like extraction at a constant volumetric rate, is approximated by a radial basis function approach which takes advantage of decomposing the original domain. The discretization error of such scheme is numerically estimated and we also face up to instability issues. This produces an effective tool for real applications, as confirmed by comparisons with classical grid-based approaches.

An Efficient Localized Meshless Method Based on the Space–Time Gaussian RBF for High-Dimensional Space Fractional Wave and Damped Equations

“…The versatility of scattered data interpolation techniques is confirmed by a lot of applications, e.g., surface reconstruction, image restoration and inpainting, meshless/Lagrangian methods for fluid dynamics, surface deformation or motion capture systems allowing the recording of sparse motions from deformable objects such as human faces and bodies [6]. The numerical solution of partial differential equations by a global collocation approach based on RBF, is also referred to as a strong form solution in the PDE literature [7][8][9][10]. An alternative interesting approach in collocation methods is to use other bases as for example Hermite exponential spline defined in [11].…”

Data-driven extrapolation via feature augmentation based on variably scaled thin plate splines

“…In this work we instead consider kernel-based meshfree models constructed via polyharmonic splines whose definition includes in particular the cubic RBF and the Thin Plate Spline (TPS); for further details see [5]. For the extrapolation issue, we take advantage of the use of the so-called Variably Scaled Kernels (VSKs) [6], which might lead to more stable and accurate schemes [7]. Here the VSKs are introduced to define a feature augmentation strategy (see e.g.…”