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Nonconforming spectral element method: a friendly introduction in one dimension and a short review in higher dimensions

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In this article we present the nonconforming spectral element method (NSEM) for linear second-order ordinary differential equations with boundary conditions having analytic solutions, and the interface problem. A fully discrete spectral element method for one-dimensional parabolic problems with smooth solutions and parabolic interface problem also have been considered. The stability and error estimates of NSEM for a two-point boundary value problem with analytic solutions are derived. The numerical formulation is described and it is essentially a least-squares formulation and spectral elements are nonconforming. The normal equations which arise in the minimization of the functional are solved using preconditioned conjugate gradient method without storing the stiffness matrix and load vector. The method is exponentially accurate. In the fully discrete formulation for the parabolic problems, Crank–Nicolson scheme is used in time variable and higher-order spectral elements are used in spatial variable. This method is second-order accurate in time and exponential accurate in spatial variable. Various numerical examples are presented to show the accuracy of the method. Finally, we review the nonconforming spectral element method for various problems in higher dimensions.

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  1. Birla Institute of Technology and Science Pilani, Hyderabad Campus, Pilani, India

    N. Kishore Kumar & Shivangi Joshi

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Corresponding author has contributed to the development of theoretical estimates and numerical algorithm. Co-author has contributed to the development of numerical code and obtained the numerical results.

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Kumar, N.K., Joshi, S. Nonconforming spectral element method: a friendly introduction in one dimension and a short review in higher dimensions. Comp. Appl. Math. 42, 139 (2023). https://doi.org/10.1007/s40314-023-02271-4

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