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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 $$\Delta^2 u+b(\bold{x}) u=f(\bold{x})\quad \text{in}\ \ \Omega,\quad \frac{\partial u}{\partial\nu}=\frac{\partial \Delta u}{\partial\nu}=0 \quad \text{on}\ \ \partial\Omega,$$ where $\Omega\subset \bbfR^d\ (d\leq 3)$ is an open and bounded domain, having a $C^1$-boundary, and the following parabolic problem $$\gather \frac{\partial u}{\partial t}+\Delta^2 u+b(\bold{x},t) u=f(\bold{x},t)\quad \text{in}\ \ \Omega\times (0,T)\\ \frac{\partial u}{\partial\nu}=\frac{\partial \Delta u}{\partial\nu}=0\quad \text{on}\ \ \partial\Omega\times (0,T),\quad u(\bold{x},0)=u_0(\bold{x}), \endgather$$ where $b(\bold{x},t), (\bold{x},t)\in\Omega\times (0,T),$ is a given function. Error estimates are given.

65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
65N15Error bounds (BVP of PDE)
65M15Error bounds (IVP of PDE)
35J40Higher order elliptic equations, boundary value problems
35K30Higher order parabolic equations, initial value problems
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
Full Text: DOI
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