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High-order numerical integration on regular embedded surfaces (2403.09178v1)
Published 14 Mar 2024 in math.NA and cs.NA
Abstract: We present a high-order surface quadrature (HOSQ) for accurately approximating regular surface integrals on closed surfaces. The initial step of our approach rests on exploiting square-squeezing--a homeomorphic bilinear square-simplex transformation, re-parametrizing any surface triangulation to a quadrilateral mesh. For each resulting quadrilateral domain we interpolate the geometry by tensor polynomials in Chebyshev--Lobatto grids. Posterior the tensor-product Clenshaw-Curtis quadrature is applied to compute the resulting integral. We demonstrate efficiency, fast runtime performance, high-order accuracy, and robustness for complex geometries.
- A. Bonito and R. H. Nochetto. Geometric Partial Differential Equations — Part I. Elsevier, 2020.
- H. Brezis. Functional analysis, sobolev spaces and partial differential equations. 2010.
- Q. Chen and I. Babuška. Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle. Computer Methods in Applied Mechanics and Engineering, 128(3):405–417, 1995.
- M. G. Duffy. Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM Journal on Numerical Analysis, 19:1260–1262, 1982.
- G. Dziuk and C. M. Elliott. Finite element methods for surface PDEs. Acta Numerica, 22:289–396, 2013.
- Estimates of the error in gauss–legendre quadrature for double integrals. Journal of Computational and Applied Mathematics, 236(6):1552–1561, 2011.
- D. Fortunato. A high-order fast direct solver for surface PDEs. arXiv preprint arXiv:2210.00022, 2022.
- Adsorption surface area and porosity. Journal of The Electrochemical Society, 114(11):279, nov 1967.
- Multivariate interpolation in unisolvent nodes–lifting the curse of dimensionality. arXiv preprint arXiv:2010.10824, 2020.
- W. Heinrichs and B. I. Loch. Spectral schemes on triangular elements. Journal of Computational Physics, 173(1):279–301, 2001.
- minterpy – Multivariate interpolation in Python. https://github.com/casus/minterpy, 2021.
- Spectral/hpℎ𝑝hpitalic_h italic_p element methods for computational fluid dynamics. Oxford University Press, 2005.
- J.-O. Lachaud. Convergent geometric estimators with digital volume and surface integrals. In Discrete Geometry for Computer Imagery, pages 3–17. Springer, 2016.
- P.-O. Persson and G. Strang. A simple mesh generator in MATLAB. SIAM Review, 46(2):329–345, 2004.
- S. Praetorius and F. Stenger. Dune-CurvedGrid – a Dune module for surface parametrization. Archive of Numerical Software, page Vol. 1 No. 1 (2022), 2022.
- J. C. Riviere and S. Myhra. Handbook of surface and interface analysis: methods for problem-solving. CRC press, 2009.
- M. Spivak. A Comprehensive Introduction to Differential Geometry, volume 1. Publish or Perish Incorporated, 1999.
- M. A. Taylor and B. Wingate. A generalized diagonal mass matrix spectral element method for non-quadrilateral elements. Applied Numerical Mathematics, 33(1-4):259–265, 2000.
- L. N. Trefethen. Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics, 2000.
- L. N. Trefethen. Approximation theory and approximation practice, volume 164. SIAM, 2019.
- S. Xiang and F. A. Bornemann. On the convergence rates of gauss and clenshaw-curtis quadrature for functions of limited regularity. SIAM J. Numer. Anal., 50:2581–2587, 2012.
- High-order integration on regular triangulated manifolds reaches super-algebraic approximation rates through cubical re-parameterizations. arXiv preprint arXiv:2311.13909, 2023.
- A note on the rate of convergence of integration schemes for closed surfaces. Computational and Applied Mathematics, 43(2):1–17, 2024.
- Q. Zhou and P. Somasundaran. Surface and Interfacial Tension: Measurement, Theory, and Applications. Surfactant Science Series, volume 119. ACS Publications, 2005.