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Convergence of unsymmetric kernel-based meshless collocation methods. (English) Zbl 1165.65066
The goal of the paper is to prove convergence of variations of the symmetric kernel-based collocation method introduced by Kansa, that approximately solves a PDE problem.
The first section is an overview concerning a class of unsymmetric discretization methods.
The second section focuses on well-posed problems and regularity. Thus for a standard Poisson boundary value problem:
\[ -\Delta u=f_\Omega \quad\text{in }\Omega, \qquad u=f_D \quad\text{on }\partial\Omega.\tag{1} \]
on a bounded domain \(\Omega\subset\mathbb R^d\) with a Dirichlet data \(f_D\) on the piecewise smooth boundary \(\partial\Omega,\) one assumes that (1) is well-posed in the sense that the solution \(u\) depends continuously on the data \(f\) of the right-hand side of (1).
Also one considers the equations as being given in strong form i.e. one assumes the solution \(u\) to be regular enough to pose the equations pointwise as:
\[ \begin{aligned} (-\Delta u)(x)=(\delta_x\circ(-\Delta))(u)=f_\Omega(x), &\quad\text{for all } x\in\Omega,\\ u(x)=(\delta_x\circ Id)(u)=f_D(x), &\quad\text{for all }x\in\partial\Omega. \end{aligned}\tag{2} \]
The third section is devoted to approximation from trial spaces. One concludes that is more convenient to use small trial spaces designed to capture essential features of the solution. Thus, the resulting linear systems get unsymmetric, because any solution from a small trial space must be tested on a fine-grained space discretization, asking for many more degrees of freedom on the “test side” than on “trial side”.
The fourth section analyses the trial spaces provided by kernels. Here the notions of translation and dilation play an important role.
The sixth section focuses on stability conditions for meshless kernel-based trial spaces.
Strong convergence and weak convergence in Soboloev spaces are discussed in the sixth and seven section, respectively.
The class of numerical techniques used to solve the discrete problem are characterized within the eight section.
The ninth section is related to the case of ill-problem, where the presented method is still useful.
The last section presents the main conclusions and future possible generalizations.

MSC:
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65F22 Ill-posedness and regularization problems in numerical linear algebra
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