Error analysis of Legendre-Galerkin spectral method for a parabolic equation with Dirichlet-Type non-local boundary conditions

• Published in: Mathematical modelling and analysis Vol. 26; no. 2; pp. 287 - 303
• Main Authors: Chattouh, Abdeldjalil, Saoudi, Khaled
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 26.05.2021
• Subjects: Boundary conditions; Boundary value problems; Differential equations, Partial; Dirichlet problem; Discretization; Error analysis; error estimate; Finite difference method; Galerkin method; Mathematical analysis; Mathematical research; non-local boundary conditions; Parabola; parabolic equation; Polynomials; Spectra; Spectral methods; Algeria; error estimate; spectral methods; Galerkin method; non-local boundary conditions; parabolic equation
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/mma.2021.12865

An efficient Legendre-Galerkin spectral method and its error analysis for a one-dimensional parabolic equation with Dirichlet-type non-local boundary conditions are presented in this paper. The spatial discretization is based on Galerkin formulation and the Legendre orthogonal polynomials, while the time derivative is discretized by using the symmetric Euler finite difference schema. The stability and convergence of the semi-discrete spectral approximation are rigorously set up by following a novel approach to overcome difficulties caused by the non-locality of the boundary condition. Several numerical tests are included to confirm the efficacy of the proposed method and to support the theoretical results.