Abstract
We develop accurate and efficient spectral methods for elliptic PDEs in complex domains using a fictitious domain approach. Two types of Petrov–Galerkin formulations with special trial and test functions are constructed, one is suitable only for the Poisson equation but with a rigorous error analysis, the other works for general elliptic equations but its analysis is not yet available. Our numerical examples demonstrate that our methods can achieve spectral convergence, i.e., the convergence rate only depends on the smoothness of the solution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adcock, B., Huybrechs, D., Martin-Vaquero, J.: On the numerical stability of fourier extensions. Found. Comput. Math. 14, 635–687 (2014)
Albin, N., Bruno, O.P.: A spectral fc solver for the compressible navier-stokes equationsin general domains i: explicit time-stepping. J. Comput. Phys. 230, 6248–6270 (2011)
Albin, N., Bruno, O.P., Cheung, T.Y., Cleveland, R.O.: Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams. J. Acoust. Soc. Am. 132, 2371–2387 (2012)
Angot, P., Pan, C.-H.B., Fabrie, P.: A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81, 497–520 (1999)
Babuska, I., Aziz, A.K.: Survey lectures on the mathematical foundation of the finite element method. In: Aziz, A.K. (ed.) The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Academic Press, New York (1972)
Boyd, J.P.: A comparison of numerical algorithms for fourier extension of the first, second, and third kinds. J. Comput. Phys. 178, 118–160 (2002)
Bruno, O.P., Lyon, M.: High-order unconditionally stable fc-ad solvers for general smooth domains i. basic elements. J. Comput. Phys. 229, 2009–2033 (2010)
Buffat, M., Le Penven, L.: A spectral fictitious domain method with internal forcing for solving elliptic pdes. J. Comput. Appl. Math. 230, 2433–2450 (2011)
Dinh, Q.V., Glowinski, R., He, J., Kwock, V., Pan, T.W., Periaux, J.: Lagrange multiplier approach to fictitious domain methods: application to fluid dynamics and electro-magnetics. In: Keyes, D.E., Chan, T.F., Meurant, G., Scroggs, J.S., Voigt, R.G. (eds.) Domain Decomposition Methods for Partial Differential Equations. SIAM, Philadelphia (1992)
Elghaoui, M., Pasquetti, R.: A spectral embedding method applied to the advection–diffusion equation. J. Comput. Phys. 125, 464–476 (1996)
Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, Berlin (2004)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)
Glowinski, R., Pan, T.-W., Periaux, J.: A fictitious domain method for dirichlet problem and applications. Comput. Methods Appl. Mech. Eng. 111, 283–303 (1994)
Lui, S.H.: Spectral domain embedding for elliptic pdes in complex domains. J. Comput. Appl. Math. 225, 541–557 (2009)
Lyon, M.: A fast algorithm for fourier continuation. SIAM J. Sci. Comput. 33, 3241–3260 (2011)
Lyon, M., Bruno, O.P.: High-order unconditionally stable fc-ad solvers for general smooth domains ii. elliptic, parabolic and hyperbolic pdes; theoretical considerations. J. Comput. Phys. 229, 3358–3381 (2010)
Le Penven, L., Buffat, M.: On the spectral accuracy of a fictitious domain method for elliptic operators in multi-dimensions. J. Comput. Phys. 231, 7893–7906 (2012)
Orszag, S.A.: Spectral methods for complex geometries. J. Comput. Phys. 37, 70–92 (1980)
Schneider, K.: Numerical simulation of the transient flow behaviour in chemical reactors using a penalisation method. Comput. Fluids 34, 1223–1238 (2005)
Shen, J.: Efficient spectral-Galerkin method I. direct solvers for second- and fourth-order equations by using Legendre polynomials. SIAM J. Sci. Comput. 15, 1489–1505 (1994)
Shen, J.: Efficient spectral-Galerkin method II. direct solvers for second- and fourth-order equations by using Chebyshev polynomialS. SIAM J. Sci. Comput. 16, 74–87 (1995)
Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)
Strang, G.: Variational crimes in the finite element method. In: Aziz, A.K. (ed.) The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Academic Press, New York (1972)
van yen, R.N., Kolomenskiy, D., Schneider, K.: Approximation of the laplace and stokes operators with dirichlet boundary conditions through volume penalization: a spectral viewpoint. Numer. Math. 128, 301–338 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported in part by NSF Grant DMS-1720442 and AFOSR Grant FA9550-16-1-0102.
Rights and permissions
About this article
Cite this article
Gu, Y., Shen, J. Accurate and Efficient Spectral Methods for Elliptic PDEs in Complex Domains. J Sci Comput 83, 42 (2020). https://doi.org/10.1007/s10915-020-01226-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-020-01226-9