Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions
Mixed methods for solving fourth-order elliptic and parabolic problems by using radial basis functions are developed. The author considers the elliptic problem of the form \[ \Delta^2 u+b(\mathbf{x}) u=f(\mathbf{x})\quad \text{in}\;\;\Omega,\quad \frac{\partial u}{\partial\nu}=\frac{\partial \Delta u}{\partial\nu}=0 \quad \text{on}\;\;\partial\Omega, \] where \(\Omega\subset \mathbb{R}^d\;(d\leq 3)\) is an open and bounded domain, having a \(C^1\)-boundary, and the following parabolic problem \[ \begin{gathered} \frac{\partial u}{\partial t}+\Delta^2 u+b(\mathbf{x},t) u=f(\mathbf{x},t)\quad \text{in}\;\;\Omega\times (0,T)\\ \frac{\partial u}{\partial\nu}=\frac{\partial \Delta u}{\partial\nu}=0\quad \text{on}\;\;\partial\Omega\times (0,T),\quad u(\mathbf{x},0)=u_0(\mathbf{x}), \end{gathered} \] where \(b(\mathbf{x},t), (\mathbf{x},t)\in\Omega\times (0,T),\) is a given function. Error estimates are given.
scientific article; zbMATH DE number 2162342
scientific article