Xiangxiong Zhang, Associate Professor, Purdue University

Yu Haijun, Professor

Recent Progress on Q^k Spectral Element Method: Accuracy, Monotonicity and Applications

10:00-11:00 August 4(Thursday), Tencent Meeting ID: 552-124-786

For implicit time discretizations, it is a much harder problem to preserve positivity. It is related to the fact that second order centered difference and piecewise linear finite element method on triangular meshes for the Laplacian operator has a monotone stiffness matrix, i.e., the inverse of the stiffness matrix has non-negative entries because the stiffness matrix is an M-matrix. Most high order schemes do not have this property. In this talk, I will present a brand new result of the finite difference implementation of continuous finite element method with tensor product of quadratic polynomial basis: it is monotone thus satisfies the discrete maximum principle for the variable coefficient Poisson equation. Such a scheme can be proven to be fourth order accurate. This is the first time that a high order accurate scheme that is proven to satisfy the discrete maximum principle for a variable coefficient diffusion operator. In the literature, spectral element methods usually refer to finite element methods with high order polynomial basis. The Q^k spectral element method has been a popular high order method for solving second order PDEs, e.g., wave equations, for more than three decades, obtained by continuous finite element method with tenor product polynomial of degree k and with at least (k+1)-point Gauss-Lobatto quadrature. The new results of this classical scheme, include its accuracy, monotonicity (stability), and examples of using monotonicity to construct high order accurate bound (or positivity) preserving schemes in various applications including the Allen-Cahn equation coupled with an incompressible velocity field, Keller-Segel equation for chemotaxis, and nonlinear eigenvalue problem for Gross–Pitaevskii equation. The application to compressible Navier-Stokes equations will be briefly explained. 1) Accuracy: when the least accurate (k+1)-point Gauss-Lobatto quadrature is used, the spectral element method is also a finite  difference (FD) scheme, and this FD scheme can sometimes be (k+2)-th order accurate for k>=2. This has been observed in practice but never proven before as rigorous a priori error estimates. We are able to prove it for linear elliptic, wave, parabolic and Schrödinger equations for Dirichlet boundary conditions. For Neumann boundary conditions, (k+2)-th order can be proven if there is no mixed second order derivative. Otherwise, only (k+3/2)-th order can be proven and some order loss is indeed observed in numerical tests. The accuracy result also applies to spectral element method on any curvilinear mesh that can be smoothly mapped to a rectangular mesh, e.g., solving a wave equation on an annulus region with a curvilinear mesh generated by polar coordinates. 2) Monotonicity: consider solving the Poisson equation, then a scheme is called monotone if the inverse of the stiffness matrix is entrywise non-negative. It is well known that second order centered difference or P1 finite element method can form an M-matrix thus they are monotone, and high order accurate schemes in general are not M-matrices and are not monotone. But there are exceptions. In particular, we have proven that the fourth order accurate FD scheme (Q^2 spectral element method) is a product of two M-matrices thus monotone for a variable coefficient diffusion operator: this is the first time that a high order accurate scheme is proven monotone for a variable coefficient operator. We have also proven the fifth order accurate FD scheme (Q^3 spectral element method) is a product of three M-matrices thus monotone for the Poisson equation: this is the first time that a fifth order accurate discrete Laplacian is proven monotone in two dimensions (all previously known high order monotone discrete Laplacian in 2D are fourth order accurate).