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On an accurate numerical integration for the triangular and tetrahedral spectral finite elements

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Abstract

In the triangular/tetrahedral spectral finite elements, we apply a bilinear/trilinear transformation to map a reference square/cube to a triangle/tetrahedron, which consequently maps the \(\varvec{Q_k}\) polynomial space on the reference element to a finite element space of rational/algebraic functions on the triangle/tetrahedron. We prove that the resulting finite element space, even under this singular referencing mapping, can retain the property of optimal-order approximation. In addition, we prove that the standard Gauss-Legendre numerical integration would provide sufficient accuracy so that the finite element solutions converge at the optimal order. In particular, the finite element method, with singular mappings and numerical integration, preserves \(\varvec{P_k}\) polynomials. That is, the \(\varvec{Q_k}\) finite element solution is exact if the true solution is a \(\varvec{P_k}\) polynomial. Numerical tests are provided, verifying all theoretic findings.

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References

  1. Arnold, D.N., Boffi, D., Falk, R.S.: Approximation by quadrilateral finite elements. Math. Comp. 71, 909–922 (2002)

    Article  MathSciNet  Google Scholar 

  2. Arunakirinathar, K., Reddy, B.D.: A stable affine-approximate finite element method. SIAM J. Numer. Anal. 40(1), 180–197 (2002)

    Article  MathSciNet  Google Scholar 

  3. Ciarlet, P.G., Raviart, P.-A.: Interpolation theory over curved elements with applications to finite element methods. Comput. Methods Appl. Mech. Engrg. 1, 217–249 (1972)

    Article  MathSciNet  Google Scholar 

  4. Durufle, M., Grob, P., Joly, P.: Influence of Gauss and Gauss-Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain. Numer. Methods Partial Differential Equations 25(3), 526–551 (2009)

    Article  MathSciNet  Google Scholar 

  5. Li, Y., Wang, L.L., Li, H., Ma, H.: A new spectral method on triangles. Lecture Notes in Computational Sciences and Engineering, Springer 76, 237–246 (2011)

    Article  MathSciNet  Google Scholar 

  6. Samson, M., Li, H., Wang, L.L.: A new triangular spectral element method I: implementation and analysis on a triangle. Numer. Algor. 64(3), 519–547 (2013)

    Article  MathSciNet  Google Scholar 

  7. Xie, Z., Zhang, S.: Accurate numerical integration for the quadrilateral and hexahedral finite elements. Numer, Methods Partial Differential Equations 37(3), 2446–2457 (2021)

    Article  MathSciNet  Google Scholar 

  8. Zhang, S.: On the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation. Numer. Math. 98(3), 559–579 (2004)

    Article  MathSciNet  Google Scholar 

  9. Zhang, S.: Numerical integration with Taylor truncations for the quadrilateral and hexahedral finite elements. J. Comput. Appl. Math. 205(1), 325–342 (2007)

    Article  MathSciNet  Google Scholar 

  10. Zhou, B., Wang, B., Wang, L., Xie, Z.: A new triangular spectral element method II: mixed formulation and hp-error estimates. Numer. Math. Theor. Meth. Appl. 12(1), 72–97 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Funding

The work of Ziqing Xie was supported in part by the National Natural Science Foundation of China (Grant Nos. 12171148 and 11771138) and the Construct Program of the Key Discipline in Hunan Province.

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Authors and Affiliations

  1. School of Mathematics, Hunan Normal University, Changsha, 410081, China

    Ziqing Xie

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

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  1. Ziqing Xie
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  2. Shangyou Zhang
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All authors made equal contribution.

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Correspondence to Shangyou Zhang.

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The submitted work is original and is not published elsewhere in any form or language. This article does not contain any studies involving animals. This article does not contain any studies involving human participants.

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Communicated by: Lothar Reichel

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Xie, Z., Zhang, S. On an accurate numerical integration for the triangular and tetrahedral spectral finite elements. Adv Comput Math 50, 63 (2024). https://doi.org/10.1007/s10444-024-10173-0

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  • DOI: https://doi.org/10.1007/s10444-024-10173-0

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