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Optimal error estimates of the Legendre tau method for second-order differential equations. (English) Zbl 1203.65118
Summary: We prove that the Legendre tau method has the optimal rate of convergence in \(L ^{2}\)-norm, \(H ^{1}\)-norm and \(H ^{2}\)-norm for one-dimensional second-order steady differential equations with three kinds of boundary conditions and in \(C([0,T];L ^{2}(I))\)-norm for the corresponding evolution equation with the Dirichlet boundary condition. For the generalized Burgers equation, we develop a Legendre tau-Chebyshev collocation method, which can also be optimally convergent in \(C([0,T];L ^{2}(I))\)-norm. Finally, we give some numerical examples.

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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