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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:
65N35Spectral, collocation and related methods (BVP of PDE)
65F05Direct methods for linear systems and matrix inversion (numerical linear algebra)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
35J40Higher order elliptic equations, boundary value problems
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)