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Efficient spectral-Galerkin method. I: Direct solvers of second- and fourth-order equations using Legendre polynomials. (English) Zbl 0811.65097
Some efficient algorithms based on the spectral Legendre-Galerkin approximations for the direct solution of second- and fourth-order elliptic partial differential equations are presented. The key to the efficiency of the suggested algorithms is the construction of appropriate basis functions, which lead to systems with sparse matrices for the discrete variational formulations.
The complexities of the algorithm are a small multiple of \(N^{d+1}\) operations for a \(d\)-dimensional domain with \((N-1)^ d\) unknowns, while the convergence rates of the algorithms are exponential for problems with smooth solutions. In addition, the algorithms can be effectively parallelized since bottlenecks of the algorithms are matrix-matrix multiplications.

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65Y20 Complexity and performance of numerical algorithms
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65F05 Direct numerical methods for linear systems and matrix inversion
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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