Paper Info
Title
Rbf Collocation And Hybrid-Lhi Methods For Stokes Systems And Its Application To Controllability Problems
Abstract
The purpose of this article is to introduce radial basis function (RBF) methods for solving both direct Stokes equations and controllability problems for the Stokes system with few internal scalar controls. In both cases, Dirichlet or Navier-slip boundary conditions are considered. We introduce two radial basis function solvers, one global and the other local, to solve Stokes equations. These methods are used to discretize the primal and adjoint systems related to the controllability problems. Both techniques are based on divergence-free global RBFs. A global colocation technique based on Div-free inverse multi-quadrics is formulated and analyzed. A generalization of scalar hybrid kernels to a vector divergence-free hybrid RBFs setting is defined. Based on these kernels, the local Hermite interpolation (LHI) method in vector form is introduced. Due to the properties of the hybrid kernel, we show that due to the properties of the hybrid kernel this local method, can reduce up to double precision, the value of the condition number of the local Gram matrices. Simultaneously, it is proved that the real components of the eigenvalues corresponding to the global LHI matrix are negative and that consequently backward difference formulas are stable for time integration. The conjugate gradient algorithm is adapted to the radial basis function setting to solve the controllability problems. Several benchmarks problems in two dimensions with a non-convex domain (a star shape) are numerically solved by these RBFs methods to display and compare their feasibility. The solutions to these problems are also implemented by finite element techniques to study their relative performance.
Year
DOI
Venue
2021
10.1007/s40314-020-01400-7
COMPUTATIONAL & APPLIED MATHEMATICS
Keywords
DocType
Volume
Stokes system, Controllability, Navier-slip boundary conditions, Radial basis functions, Local Hermite interpolation method, Hybrid kernels
Journal
40
Issue
ISSN
Citations 
1
2238-3603
0
PageRank 
References 
Authors
0.34
0
3
Name
Order
Citations
PageRank
Louis Breton100.34
Pedro González-Casanova200.68
Cristhian Montoya300.34