Abstract
A polar coordinate transformation is considered, which transforms the complex geometries into a unit disc. Some basic properties of the polar coordinate transformation are given. As applications, we consider the elliptic equation in two-dimensional complex geometries. The existence and uniqueness of the weak solution are proved, the Fourier–Legendre spectral-Galerkin scheme is constructed and the optimal convergence of numerical solutions under \(H^1\)-norm is analyzed. The proposed method is very effective and easy to implement for problems in 2D complex geometries. Numerical results are presented to demonstrate the high accuracy of our spectral-Galerkin method.
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References
An, J., Li, H.Y., Zhang, Z.M.: Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains. Numer. Algorithms 84, 427–455 (2020)
Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Springer, Berlin (2001)
Buzbee, B.L., Dorr, F.W., George, J.A., Golub, G.H.: The direct solution of the discrete Poisson equation on irregular regions. SIAM J. Numer. Anal. 8, 722–736 (1971)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Fundamentals in Single Domains. Springer, Berlin (2006)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, Berlin (2007)
Gordon, W.J., Hall, C.A.: Transfinite element methods: blending-function interpolation over arbitrary curved element domains. Numer. Math. 21, 109–129 (1973)
Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia (1977)
Gu, Y.Q., Shen, J.: Accurate and efficient spectral methods for elliptic PDEs in complex domains. J. Sci. Comput. 83, 42, 20 (2020)
Gu, Y.Q., Shen, J.: An efficient spectral method for elliptic PDEs in complex domains with circular embedding. SIAM J. Sci. Comput. 43, A309–A329 (2021)
Guo, B.Y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)
Heinrichs, W.: Spectral collocation schemes on the unit disc. J. Comput. Phys. 199, 66–86 (2004)
Karniadakis, G., Sherwin, S.: Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2005)
Korczak, K.Z., Patera, A.T.: An isoparametric spectral element method for solution of the Navier–Stokes equations in complex geometry. J. Comput. Phys. 62, 361–382 (1986)
Lui, S.H.: Spectral domain embedding for elliptic PDEs in complex domains. J. Comput. Appl. Math. 225, 541–557 (2009)
Orszag, S.A.: Spectral methods for problems in complex geometries. J. Comput. Phys. 37, 70–92 (1980)
Peyret, R.: Spectral Methods for Incompressible Viscous Flow. Springer, New York (2002)
Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford (1999)
Shen, J.: Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials. SIAM J. Sci. Comput. 15, 1489–1505 (1994)
Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)
Shen, J., Wang, L.L., Li, H.Y.: A triangular spectral element method using fully tensorial rational basis functions. SIAM J. Numer. Anal. 47, 1619–1650 (2009)
Song, F.Y., Xu, C.J., Karniadakis, G.E.: Computing fractional Laplacians on complex-geometry domains: algorithms and simulations. SIAM J. Sci. Comput. 39, A1320–A1344 (2017)
Toselli, A., Widlund, O.: Domain Decomposition Methods-Algorithms and Theory. Springer, Berlin (2005)
Yi, L.J., Guo, B.Q.: An h-p version of the continuous Petrov–Galerkin finite element method for Volterra integro-differential equations with smooth and nonsmooth kernels. SIAM J. Numer. Anal. 53, 2677–2704 (2015)
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The work of these authors is supported by the National Natural Science Foundation of China (No. 12071294) and Natural Science Foundation of Shanghai (No. 22ZR1443800).
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Z. Wang, X. Wen and G. Yao all contributed to the conceptualization, methodology, writing and the numerical simulations.
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The work of these authors is supported by the National Natural Science Foundation of China (No. 12071294) and Natural Science Foundation of Shanghai (No. 22ZR1443800).
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Wang, Z., Wen, X. & Yao, G. An Efficient Spectral-Galerkin Method for Elliptic Equations in 2D Complex Geometries. J Sci Comput 95, 89 (2023). https://doi.org/10.1007/s10915-023-02207-4
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DOI: https://doi.org/10.1007/s10915-023-02207-4
Keywords
- PDEs in complex geometries
- Polar coordinate transformation
- Fourier–Legendre spectral-Galerkin methods
- Convergence
- Numerical results