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Accuracy and Monotonicity of Spectral Element Method on Structured Meshes
thesisposted on 03.05.2021, 19:40 by Hao Li
On rectangular meshes, the simplest spectral element method for elliptic equations is the classical Lagrangian Qk finite element method with only (k+1)-point Gauss-Lobatto quadrature, which can also be regarded as a finite difference scheme on all Gauss-Lobatto points. We prove that this finite difference scheme is (k + 2)-th order accurate for k ≥ 2, whereas Qk spectral element method is usually considered as a (k + 1)-th order accurate scheme in L2-norm. This result can be extended to linear wave, parabolic and linear Schrödinger equations.
Additionally, the Qk finite element method for elliptic problems can also be viewed as a finite difference scheme on all Gauss-Lobatto points if the variable coefficients are replaced by their piecewise Qk Lagrange interpolants at the Gauss Lobatto points in each rectangular cell, which is also proven to be (k + 2)-th order accurate.
Moreover, the monotonicity and discrete maximum principle can be proven for the fourth order accurate Q2 scheme for solving a variable coefficient Poisson equation, which is the first monotone and high order accurate scheme for a variable coefficient elliptic operator.
Last but not the least, we proved that certain high order accurate compact finite difference methods for convection diffusion problems satisfy weak monotonicity. Then a simple limiter can be designed to enforce the bound-preserving property when solving convection diffusion equations without losing conservation and high order accuracy.