Coupled Legendre Chebyshev Collocation Least Squares Method For Partial Differential Equations

The finite difference method, the finite element method, the finite volume method, and the spectral method are all effective numerical methods for solving the partial differ-ential equation. In recent years, the least squares finite element method has also caused the attention of some scholars and is investigated in. To obtain high order spec-tral accuracy, the least squares spectral method also considered by some scholars. In this paper, we investigate the Legendre-Galerkin Chebyshev collocation (LGCC) least squares method for the partial differential equation. Firstly, the original problem can be rewritten as an equivalent first-order system by introducing a flux, the LGCC least squares scheme can be developed for the system. The theoretical analysis is also con-sidered, and some numerical examples can be given to verify the spectral accuracy. The paper can be summarized as the following several parts.Part 1, we consider the one-dimensional partial differential equation. Firstly, we develop the LGCC least squares method for the two-point boundary value problems with variable coefficients. The coercivily, the continuity, and the error estimate of the scheme are derived. Secondly, the multi-domain LGCC least squares method also developed, and the implementation of the scheme in parallel is also discussed and the corresponding nu-merical theoretical analysis for the single domain is also extended. Then, we develop the multi-domain LGCC method for the parabolic equation with two nonhomogeneous jump conditions. The scheme treats the first jump condition essentially and the second one nat-urally. The proposed method is applied to the computation of the one-dimensional two phase Stefan problem. Finally, we also consider the multi-domain LGCC least squares method for the two point boundary problem with discontinuous variable coefficients and present the corresponding numerical analysis.Part 2, for two-dimensional elliptic equation, we first investigate the LGCC least squares method for the elliptic problems with variable coefficients. We present the sta-bility of the two dimension Chebyshev interpolation and the corresponding results of approximation. The results lead to the ellipticity and the continuity of our scheme, and the error estimate in the H1-norm is also derived. Secondly, the proposed method is applied to solving the Stokes equation and the corresponding numerical analysis is also given. Then, we develop the multi-domain LGCC scheme for the two dimensional el-liptic equation with two jump conditions and its implementation in parallel is designed. Finally, we also develop the multi-domain LGCC least squares method for the variable coefficients elliptic equation with two jump conditions and the parallel algorithm of the scheme is discussed. We extended the previous results about the stability of the Cheby- shev interpolation for the smooth function and the corresponding numerical theoretical results.Part 3, we investigate the Legendre-Galerkin with numerical integration (LG-NI) triangular element least squares method for the variable coefficient elliptic problems in a triangle. The corresponding coercivity, the continuity, and the convergence of the scheme are also considered. Further, combining the interior domain by using a rectangular sub-division with the border with appropriate triangle element subdivision, the multi-domain LG-NI least squares method is developed for the elliptic equation in a polygon domain and its convergence is also obtained.At the last part, we discuss the LGCC least squares method for the evolution equa-tion. For parabolic equations, the LGCC least squares method and its multi-step version are investigated, and its application to the Burgers equation is also given. Further, we discuss the LGCC least squares method for the two kind of nonlinear evolution equation. For the Burgers equation, we apply the Crank-Nicolson scheme to the time advancing for the first order system of the Burgers equations. The LGCC least squares method is adopted to the discretization of the spatial space. Another, the proposed method can be applied to two dimensional nonlinear parabolic equations.The presented method based on the Legendre-Galerkin least squares method, but the variable coefficients term can be treated by the Chebyshev collocation. The scheme leads to to a symmetric and positive definite system. Note that the proposed method inherits the stability of the Legendre method and avoids the singularity of the Chebyshev weight which arises on the interface in domain decomposition. Thus, the present method can be extended to multidomain cases and nonlinear problems.