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A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations. (English) Zbl 1170.65099

The authors develop some efficient spectral algorithms based on Jacobi-Galerkin methods for the solution of integrated forms of fourth-order differential equations in one and two variables. The spatial approximation is based on Jacobi polynomials P n (α,β) (x), with α,β(-1,) and n the polynomial degree. For α=β , one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α=β=1 2,α=β=0, the Chebyshev polynomials of the first and second kinds and Legendre polynomials, respectively. For the nonsymmetric Jacobi polynomials, the two important special cases α=-β=±1 2 (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two-dimensional version of the approximations is obtained by tensor products of the one-dimensional bases.

The resulting discrete systems have specially structured matrices that can be efficiently inverted. An algebraic preconditioning yields a condition number of O(N), (N being the the number of retained modes of approximations) which is an improvement with respect to the well-known condition number O(N 8 ) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to N d+1 for a d-dimensional problem. Numerical results are presented in which the usual exponential behaviour of spectral approximations is exhibited.

MSC:
65N35Spectral, collocation and related methods (BVP of PDE)
35J40Higher order elliptic equations, boundary value problems
65F35Matrix norms, conditioning, scaling (numerical linear algebra)
65Y20Complexity and performance of numerical algorithms