Error estimate of a Legendre-Galerkin Chebyshev collocation method for a class of parabolic inverse problem. (English) Zbl 1537.65121
Summary: A Legendre-Galerkin Chebyshev collocation method is presented for the parabolic inverse problem with control parameters. Optimal order of convergence of the semi-discrete method is obtained in \(L^2\)-norm for the nonlinear term being not globally Lipschitz continuous. For time-discretization, a Legendre-tau method is applied. The method is implemented by the explicit-implicit iterative method. Suitable basis functions are constructed leading to sparse matrices, and the nonlinear term is collocated at the Chebyshev-Gauss-Lobatto points computed explicitly by the fast Legendre transform. Numerical results are given to show the efficiency and capability of this space-time spectral method.
MSC:
| 65M32 | Numerical methods for inverse problems 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 |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65D05 | Numerical interpolation |
| 65D30 | Numerical integration |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 35K05 | Heat equation |
| 49M25 | Discrete approximations in optimal control |
Keywords:
error estimate; spectral method; Legendre-Galerkin Chebyshev collocation; inverse problem; over-specificationReferences:
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