Accuracy and Monotonicity of Spectral Element Method on Structured Meshes
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 quadra-ture, 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 k ≥ 2, whereas Qk spectral element method is usually considered as a(k+ 1)-th order accurate schemeinL2-norm. This result can be extended to linear wave, parabolic and linear Schrödingerequations. 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.
Zhang, Purdue University.
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