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**Efficient spectral-Galerkin method. II: Direct solvers of second- and fourth-order equations using Chebyshev polynomials.**
*(English)*
Zbl 0840.65113

The paper is the second in the series for developing efficient spectral-Galerkin methods for elliptic equations. In the first part [ibid. 15, No. 6, 1489-1505 (1994; Zbl 0811.65097)] direct solution techniques for the Helmholtz equation and the biharmonic equation using Legendre-Galerkin approximations have been described.

In this second part direct solvers based on Chebyshev-Galerkin methods for second-order and fourth-order equations are presented. They are based on appropriate basis functions for the Galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted. Numerical results indicate that the direct solvers presented in this paper are significantly more accurate than those based on the Chebyshev \(\tau\) method.

In this second part direct solvers based on Chebyshev-Galerkin methods for second-order and fourth-order equations are presented. They are based on appropriate basis functions for the Galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted. Numerical results indicate that the direct solvers presented in this paper are significantly more accurate than those based on the Chebyshev \(\tau\) method.

Reviewer: J.Vaníček (Praha)

### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65F05 | Direct numerical methods for linear systems and matrix inversion |

31A30 | Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions |

35J40 | Boundary value problems for higher-order elliptic equations |

35J25 | Boundary value problems for second-order elliptic equations |