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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.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J40 Boundary value problems for higher-order elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling
65Y20 Complexity and performance of numerical algorithms
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