Chebyshev Spectral Method For 3D Parabolic Equations

The purpose of this paper is to consider spectral numerical approximations to three-dimensional parabolic equations with initial boundary value problem ut-△u=f,x∈Ω,t∈(0,T), u(x,t)=0,x∈(?)Ω,t∈(0,T), u(x,0)=u0(x),x∈Ω.The bilinear function of pseudo spectral Chebyshev method for three-dimensional parabolic equations is given, and some properties of bilinear functional is presented, auch as boundedness, positive definiteness and so on. Then, we give the several operators of Fourier system and Chebyshev system with their error estimations.The semi-discretization based on pseudo spectral Chebyshev method is pre-sented. First, we give the relevant variational problem. The stability and the con-vergence of the solution is presented by using several operators of Chebyshev system and the relation between the continuous and the discrete inner products. Then, the error estimate of the scheme is obtained.The full-discretization based on pseudo spectral Chebyshev method is presented. First, we give the approximation scheme. The stability and the convergence of the solution is presented by using several operators of Chebyshev system and the relation between the continuous and the discrete inner products. Then, the error estimate of the scheme is obtained.