Finite element method for two-dimensional linear advection equations based on spline method

Kai Qu; Qi Dong; Chanjie Li; Feiyu Zhang

doi:10.3934/dcdss.2021056

Article Contents

2021, Volume 14, Issue 7: 2471-2485. Doi: 10.3934/dcdss.2021056

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Finite element method for two-dimensional linear advection equations based on spline method

College of Science, Dalian Maritime University, Dalian, 116026, China

^* Corresponding author: Kai Qu
^* Corresponding author: Kai Qu

Received: February 2020

Revised: October 2020

Early access: May 2021

Published: July 2021

The first author is supported by National Natural Science Foundation of China grant 11801053, the Fundamental Research Funds for the Central Universities (3132019176, 3132019323).

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Abstract

A new method for some advection equations is derived and analyzed, where the finite element method is constructed by using spline. A proper spline subspace is discussed for satisfying boundary conditions. Meanwhile, in order to get more accuracy solutions, spline method is connected with finite element method. Furthermore, the stability and convergence are discussed rigorously. Two numerical experiments are also presented to verify the theoretical analysis.

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Mathematics Subject Classification: Primary: 65D07, 65M60; Secondary: 41A15.

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Figure 1. Uniform type-2 triangulation, m = 4, n = 4

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Figure 2. A locally supported spline

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Figure 3. (a) Corner B-spline Basis (b)Side B-spline Basis Interior (c)B-spline Basis

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Figure 4. $ \varepsilon = 1 $, (a)$ u(0.1) $, (b)$ \hat{u}(0.1) $, (c)$ \tilde{u}(0.1) $

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Figure 5. $ \varepsilon = 1 $, (a)$ u(0.3) $, (b)$ \hat{u}(0.3) $, (c)$ \tilde{u}(0.3) $

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Figure 6. $ \varepsilon = 1 $, (a)$ u(0.5) $, (b)$ \hat{u}(0.5) $, (c)$ \tilde{u}(0.5) $

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Figure 7. $ \varepsilon = 1 $, (a)$ u(0.7) $, (b)$ \hat{u}(0.7) $, (c)$ \tilde{u}(0.7) $

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Figure 8. $ \varepsilon = 2 $, (a)$ u(0.1) $, (b)$ \hat{u}(0.1) $, (c)$ \tilde{u}(0.1) $

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Figure 9. $ \varepsilon = 2 $, (a)$ u(0.3) $, (b)$ \hat{u}(0.3) $, (c)$ \tilde{u}(0.3) $

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Figure 10. $ \varepsilon = 2 $, (a)$ u(0.5) $, (b)$ \hat{u}(0.5) $, (c)$ \tilde{u}(0.5) $

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Figure 11. $ \varepsilon = 2 $, (a)$ u(0.7) $, (b)$ \hat{u}(0.7) $, (c)$ \tilde{u}(0.7) $

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Figure 12. (a) $ u(0.1) $, (b)$ \hat{u}(0.1) $, (c)$ \tilde{u}(0.1) $

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Figure 13. (a) $ u(0.2) $, (b)$ \hat{u}(0.2) $, (c)$ \tilde{u}(0.2) $

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Figure 14. (a) $ u(0.3) $, (b)$ \hat{u}(0.3) $, (c)$ \tilde{u}(0.3) $

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Figure 15. (a) $ u(0.4) $, (b)$ \hat{u}(0.4) $, (c)$ \tilde{u}(0.4) $

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Figure 16. (a) $ u(0.5) $, (b)$ \hat{u}(0.5) $, (c)$ \tilde{u}(0.5) $

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Table 1. Comparison of numerical and exact solutions of Example 1

		Spline method	Finite element method
$ \varepsilon=1 $	$ t=0.1 $	3.774902e-005	6.416033e-005
$ \varepsilon=1 $	$ t=0.3 $	2.721077e-005	2.347618e-004
$ \varepsilon=1 $	$ t=0.5 $	6.327942e-005	7.128474e-004
$ \varepsilon=1 $	$ t=0.7 $	6.323704e-005	2.739811e-004
$ \varepsilon=2 $	$ t=0.1 $	3.573924e-004	2.159467e-003
$ \varepsilon=2 $	$ t=0.3 $	5.858324e-004	4.492941e-003
$ \varepsilon=2 $	$ t=0.5 $	8.340485e-004	6.032486e-003
$ \varepsilon=2 $	$ t=0.7 $	7.448253e-004	5.265392e-003

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Table 2. Comparison of numerical and exact solutions of Example 2

	Spline method	Finite element method
$ t=0.1 $	4.537264e-006	5.276482e-005
$ t=0.2 $	4.822438e-006	4.653391e-005
$ t=0.3 $	5.645882e-006	5.283764e-005
$ t=0.4 $	5.326159e-006	7.563822e-005
$ t=0.5 $	7.438625e-006	6.435764e-005

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