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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 $P_n^{(\alpha ,\beta )}(x)$, with $\alpha ,\beta \in (-1,\infty )$ and $n$ the polynomial degree. For $\alpha =\beta $ , one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for $\alpha =\beta =\mp \frac 12,\alpha =\beta =0$, the Chebyshev polynomials of the first and second kinds and Legendre polynomials, respectively. For the nonsymmetric Jacobi polynomials, the two important special cases $\alpha =-\beta =\pm \frac 12$ (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.

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
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