<div>On rectangular meshes, the simplest spectral element method for elliptic equations is the classical Lagrangian <i>Q</i><sup>k</sup> finite element method with only (<i>k</i>+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 (<i>k</i> + 2)-th order accurate for <i>k</i> ≥ 2, whereas <i>Q</i><sup><i>k</i></sup> spectral element method is usually considered as a (<i>k</i> + 1)-th order accurate scheme in <i>L<sup>2</sup></i>-norm. This result can be extended to linear wave, parabolic and linear Schrödinger equations.</div><div><br></div><div><div>Additionally, the <i>Q<sup>k</sup></i> 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 <i>Q<sup>k</sup> </i>Lagrange interpolants at the Gauss Lobatto points in each rectangular cell, which is also proven to be (<i>k</i> + 2)-th order accurate.</div></div><div><br></div><div><div>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.</div></div><div><br></div><div><div>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.</div><div><br></div></div>
| Identifer | oai:union.ndltd.org:purdue.edu/oai:figshare.com:article/14529456 |
| Date | 03 May 2021 |
| Creators | Hao Li (10731936) |
| Source Sets | Purdue University |
| Detected Language | English |
| Type | Text, Thesis |
| Rights | CC BY 4.0 |
| Relation | https://figshare.com/articles/thesis/Accuracy_and_Monotonicity_of_Spectral_Element_Method_on_Structured_Meshes/14529456 |
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