## Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials.(English)Zbl 1152.65112

The authors consider some 1D and 2D fourth order differential equations supplied with first order (clamped) homogeneous and nonhomogeneous boundary conditions. They solve these problems by a Galerkin type method based on finite dimensional trial and test spaces spanned on Jacobi polynomials. In fact, they introduce two distinct bases of shape functions trying to minimize the bandwidth and condition number of the coefficient matrices. These special structured discretization matrices can also be efficiently inverted. Two fairly relevant examples are carried out.

### MSC:

 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 65F05 Direct numerical methods for linear systems and matrix inversion 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35J40 Boundary value problems for higher-order elliptic equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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