*(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 ${P}_{n}^{(\alpha ,\beta )}\left(x\right)$, with $\alpha ,\beta \in (-1,\infty )$ and $n$ the polynomial degree. For $\alpha =\beta $ , one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for $\alpha =\beta =\mp \frac{1}{2},\alpha =\beta =0$, the Chebyshev polynomials of the first and second kinds and Legendre polynomials, respectively. For the nonsymmetric Jacobi polynomials, the two important special cases $\alpha =-\beta =\pm \frac{1}{2}$ (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 $O\left(N\right),$ ($N$ being the the number of retained modes of approximations) which is an improvement with respect to the well-known condition number $O\left({N}^{8}\right)$ 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 (BVP of PDE) |

35J40 | Higher order elliptic equations, boundary value problems |

65F35 | Matrix norms, conditioning, scaling (numerical linear algebra) |

65Y20 | Complexity and performance of numerical algorithms |