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Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions. (English) Zbl 1069.65107

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

${{\Delta }}^{2}u+b\left(𝐱\right)u=f\left(𝐱\right)\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\Omega },\phantom{\rule{1.em}{0ex}}\frac{\partial u}{\partial \nu }=\frac{\partial {\Delta }u}{\partial \nu }=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\partial {\Omega },$

where ${\Omega }\subset {ℝ}^{d}\phantom{\rule{4pt}{0ex}}\left(d\le 3\right)$ is an open and bounded domain, having a ${C}^{1}$-boundary, and the following parabolic problem

$\begin{array}{c}\frac{\partial u}{\partial t}+{{\Delta }}^{2}u+b\left(𝐱,t\right)u=f\left(𝐱,t\right)\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\Omega }×\left(0,T\right)\\ \frac{\partial u}{\partial \nu }=\frac{\partial {\Delta }u}{\partial \nu }=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\partial {\Omega }×\left(0,T\right),\phantom{\rule{1.em}{0ex}}u\left(𝐱,0\right)={u}_{0}\left(𝐱\right),\end{array}$

where $b\left(𝐱,t\right),\left(𝐱,t\right)\in {\Omega }×\left(0,T\right),$ is a given function. Error estimates are given.

##### MSC:
 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 65N15 Error bounds (BVP of PDE) 65M15 Error bounds (IVP of PDE) 35J40 Higher order elliptic equations, boundary value problems 35K30 Higher order parabolic equations, initial value problems 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)