Solving Interface Problems of the Helmholtz Equation by Immersed Finite Element Methods

doi:10.1007/s42967-019-0002-2

摘要/Abstract

摘要： This article reports our explorations for solving interface problems of the Helmholtz equation by immersed fnite elements (IFE) on interface independent meshes. Two IFE methods are investigated:the partially penalized IFE (PPIFE) and discontinuous Galerkin IFE (DGIFE) methods. Optimal convergence rates are observed for these IFE methods once the mesh size is smaller than the optimal mesh size which is mainly dictated by the wave number. Numerical experiments also suggest that higher degree IFE methods are advantageous because of their larger optimal mesh size and higher convergence rates.

关键词: Helmholtz interface problems, Immersed fnite element(IFE)methods, Higher degree fnite element methods

Abstract: This article reports our explorations for solving interface problems of the Helmholtz equation by immersed fnite elements (IFE) on interface independent meshes. Two IFE methods are investigated:the partially penalized IFE (PPIFE) and discontinuous Galerkin IFE (DGIFE) methods. Optimal convergence rates are observed for these IFE methods once the mesh size is smaller than the optimal mesh size which is mainly dictated by the wave number. Numerical experiments also suggest that higher degree IFE methods are advantageous because of their larger optimal mesh size and higher convergence rates.

Key words: Helmholtz interface problems, Immersed fnite element(IFE)methods, Higher degree fnite element methods

中图分类号:

Tao Lin, Yanping Lin, Qiao Zhuang. Solving Interface Problems of the Helmholtz Equation by Immersed Finite Element Methods[J]. Communications on Applied Mathematics and Computation, 2019, 1(2): 187-206.

参考文献

1. Adjerid, S., Ben-Romdhane, M., Lin, T.:Higher degree immersed fnite element methods for second-order elliptic interface problems. Int. J. Numer. Anal. Model. 11(3), 541-566(2014)
2. Adjerid, S., Ben-Romdhane, M., Lin, T.:Higher degree immersed fnite element spaces constructed according to the actual interface. Comput. Math. Appl. 75(6), 1868-1881(2018)
3. Adjerid, S., Guo, R., Lin, T.:High degree immersed fnite element spaces by a least squares method. Int. J. Numer. Anal. Model. 14(4/5), 604-626(2017)
4. Ainsworth, M.:Dispersive and dissipative behaviour of high order discontinuous Galerkin fnite element methods. J. Comput. Phys. 198, 106-130(2004)
5. Annavarapu, C., Hautefeuille, M., Dolbow, J.:A fnite element method for crack growth without remeshing. Comput. Methods Appl. Mech. Eng. 225-228, 44-54(2012)
6. Aziz, A.K., Werschulz, A.:On the numerical solutions of Helmholtz's equation by the fnite element method. SIAM J. Numer. Anal. 19(5), 166-178(1995)
7. Babuška, I.:The fnite element method for elliptic equations with discontinuous coefcients. Computing (Arch. Elektron. Rechnen) 5, 207-213(1970)
8. Babuška, I.M., Sauter, S.A.:Is the pollution efect of the FEM avoidable for the Helmholtz equation considering high wave numbers? Comput. Math. Appl. 34(6), 2392-2423(1997)
9. Barrett, J.W., Elliott, C.M.:Fitted and unftted fnite-element methods for elliptic equations with smooth interfaces. IMA J. Numer. Anal. 7(3), 283-300(1987)
10. Bonnet-Ben Dhia, A.S., Ciarlet Jr., P., Zwölf, C.M.:Time harmonic wave difraction problems in materials with sign-shifting coefcients. J. Comput. Appl. Math. 234, 1912-1919(2010)
11. Braess, D.:Finite Elements:Theory, Fast Solvers, and Applications in Solid Mechanics, 2nd edn. Cambridge University Press, Cambridge (2001). Translated from the 1992 German edition by Larry L. Schumaker
12. Bramble, J.H., King, J.T.:A fnite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6(2), 109-138(1996)
13. Brown, D.L.:A note on the numerical solution of the wave equation with piecewise smooth coeffcients. Math. Comput. 42(166), 369-391(1984)
14. Burman, E., Wu, H., Zhu, L.:Linear continuous interior penalty fnite element method for Helmholtz equation with high wave number:one-dimensional analysis. Numer. Methods Partial Difer. Equ. 32(5), 1378-1410(2016)
15. Bériot, H., Prinn, A., Gabard, G.:Efcient implementation of high-order fnite elements for Helmholtz problems. Int. J. Numer. Methods Eng. 106, 213-240(2016)
16. Chandler-wilde, S.N., Zhang, B.:Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers. SIAM J. Math. Anal. 30(3), 559-583(1989)
17. Chen, Z., Zou, J.:Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79(2), 175-202(1998)
18. Christiansen, P.S., Krenk, S.:A recursive fnite element technique for acoustic felds in pipes with absorption. J. Sound Vib. 122(1), 107-118(1988)
19. Chu, C.-C., Graham, I.G., Hou, T.-Y.:A new multiscale fnite element method for high-contrast elliptic interface problems. Math. Comput. 79(272), 1915-1955(2010)
20. Clough, R.W., Tocher, J.L.:Finite element stifness matrices for analysis of plate bending. In:Przemieniecki, J.S., Bader,R.M., Bozich, W.F., Johnson, J.R., Mykytow, W.J. (eds.) Matrix Methods in Structural Mechanics, The Proceedings of the Conference held at Wright-Parrterson Air Force Base, Ohio, 26-28, October, 1965, pp. 515-545, Washington, 1966. Air Force Flight Dynamics Laboratory
21. Douglas Jr., J., Sheen, D., Santos, J.E.:Approximation of scalar waves in the space-frequency domain. Math. Models Methods Appl. Sci. 4(4), 509-531(1994)
22. Du, Y., Wu, H.:Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number. SIAM J. Numer. Anal. 53(2), 782-804(2015)
23. Farhat, C., Harari, I., Hetmaniuk, U.:A discontinuous Galerkin method with lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Eng. 192, 1389-1419(2003)
24. Feng, X., Wu, H.:Discontinuous Galerkin methods for the Helmholtz equation with large wave number. SIAM J. Numer. Anal. 47(4), 2872-2896(2009)
25. Gittelson, G., Hiptmair, R., Perugia, I.:Plane wave discontinuous Galerkin methods:analysis of the h-version. Esaim Math. Model. Numer. Anal. 43, 297-331(2009)
26. He, X., Lin, T., Lin, Y.:Approximation capability of a bilinear immersed fnite element space. Numer. Methods Partial Difer. Equ. 24(5), 1265-1300(2008)
27. He, X., Lin, T., Lin, Y.:Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefcient. J. Syst. Sci. Complex. 23(3), 467-483(2010)
28. He, X., Lin, T., Lin, Y.:Immersed fnite element methods for elliptic interface problems with nonhomogeneous jump conditions. Int. J. Numer. Anal. Model. 8(2), 284-301(2011)
29. He, X., Lin, T., Lin, Y.:The convergence of the bilinear and linear immersed fnite element solutions to interface problems. Numer. Methods Partial Difer. Equ. 28(1), 312-330(2012)
30. He, X., Lin, T., Lin, Y.:A selective immersed discontinuous Galerkin method for elliptic interface problems. Math. Methods Appl. Sci. 37(7), 983-1002(2014)
31. He, X.:Bilinear immersed fnite elements for interface problems. PhD thesis, Virginia Polytechnic Institute and State University (2009)
32. Hou, T.Y., Wu, X.:A multiscale fnite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169-189(1997)
33. Ihlenburg, F., Babuška, I.:Finite element solution of the Helmholtz equation with high wave number. I. The h-version of the FEM. Comput. Math. Appl. 30(9), 9-37(1995)
34. Ihlenburg, F., Babuška, I.:Finite element solutions of the Helmholtz equation with high wave number part Ⅱ:the h-p version of the FEM. SIAM J. Numer. Anal. 34(1), 315-358(1997)
35. Jensen, F.B., Kuperman, W.A., Porter, M.B., Schmidt, H.:Computational Ocean Acoustics. Springer, Berlin (1995)
36. Klenow, B., Nisewonger, A., Batra, R.C., Brown, A.:Refection and transmission of plane waves at an interface between two fuids. Comput. Fluids 36, 1298-1306(2007)
37. Kreiss, H., Petersson, N.A.:An embedded boundary method for the wave equation with discontinuous coefcients. SIAM J. Sci. Comput. 28(6), 2054-2074(2006)
38. Lam, C.Y., Shu, C.-W.:A phase-based interior penalty discontinuous Galerkin method for the Helmholtz equation with spatially varying wavenumber. Comput. Methods Appl. Mech. Eng. 318, 456-473(2017)
39. LeVeque, R.J., Li, Z.L.:The immersed interface method for elliptic equations with discontinuous coeffcients and singular sources. SIAM J. Numer. Anal. 31(4), 1019-1044(1994)
40. Li, Z., Ito, K.:The Immersed Interface Method:Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. Frontiers in Applied Mathematics, vol. 33. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)
41. Li, Z., Lin, T., Lin, Y., Rogers, R.C.:An immersed fnite element space and its approximation capability. Numer. Methods Partial Difer. Equ. 20(3), 338-367(2004)
42. Li, Z., Lin, T., Wu, X.:New Cartesian grid methods for interface problems using the fnite element formulation. Numer. Math. 96(1), 61-98(2003)
43. Lin, T., Lin, Y., Rogers, R., Lynne Ryan, M.:A Rectangular Immersed Finite Element Space for Interface Problems. Scientifc Computing and Applications (Kananaskis, AB, 2000). Advances in Computation:Theory and Practice, vol. 7, pp. 107-114. Nova Sci. Publ, Huntington (2001)
44. Lin, T., Lin, Y., Zhang, X.:Partially penalized immersed fnite element methods for elliptic interface problems. SIAM J. Numer. Anal. 53(2), 1121-1144(2015)
45. Lin, T., Yang, Q., Zhang, X.:A priori error estimates for some discontinuous Galerkin immersed fnite element methods. J. Sci. Comput. 65, 875-894(2015)
46. Parsania, A., Melenk, J.M., Sauter, D.:General DG-methods for highly indefnite Helmholtz problems. J. Sci. Comput. 57, 536-581(2013)
47. Perugia, I.:A note on the discontinuous Galerkin approximation of the Helmholtz equation. Lecture notes. ETH, Zürich (2006)
48. Romdhane, M.B.:Higher-degree immersed fnite elements for second-order elliptic interface problems. PhD thesis, Virginia Polytechnic Institute and State University (2011)
49. Semblat, J.F., Brioist, J.J.:Efciency of higher order fnite elements for the analysis of seismic wave propagation. J. Sound Vib. 231(2), 460-467(2000)
50. Speck, F.O.:Sommerfeld difraction problems with frst and second kind boundary conditions. SIAM J. Math. Anal. 20(2), 396-407(1989)
51. Suater, S.A., Warnke, R.:Composite fnite elements for elliptic boudnary value problems with discontinuous coefcients. Computing 77, 29-55(2006)
52. Wang, K., Wong, Y.:Pollution-free fnite diference schemes for non-homogeneous Helmholtz equation. Int. J. Numer. Anal. Model. 11(4), 787-815(2014)
53. Xu, J.:Estimate of the convergence rate of the fnite element solutions to elliptic equation of second order with discontinuous coefcients. Nat. Sci. J. Xiangtan Univ. 1, 1-5(1982)
54. Zhang, J.:Wave propagation across fuid-solid interfaces:a grid method approach. Geophys. J. Int. 159, 240-252(2004)
55. Zhang, S.M., Li, Z.:An augmented ⅡM for Helmholtz/Poisson equations on irregular domains in complex space. Int. J. Numer. Anal. Model. 13(1), 166-178(2016)
56. Zhang, X.:Nonconforming immersed fnite element methods for interface problems. PhD thesis, Virginia Polytechnic Institute and State University (2013)
57. Zou, Z., Aquino, W., Harari, I.:Nitsche's method for Helmholtz problems with embedded interfaces. Int. J. Numer. Meth. Eng. 110, 618-636(2017)

[1]	Xiangcheng Zheng, V. J. Ervin, Hong Wang. An Indirect Finite Element Method for Variable-Coefcient Space-Fractional Difusion Equations and Its Optimal-Order Error Estimates[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 147-162.
[2]	Paola Antonietti, Claudio Canuto, Marco Verani. An Adaptive hp-DG-FE Method for Elliptic Problems: Convergence and Optimality in the 1D Case[J]. Communications on Applied Mathematics and Computation, 2019, 1(3): 309-331.
[3]	Paola Gervasio, Alfio Quarteroni. The INTERNODES Method for Non-conforming Discretizations of PDEs[J]. Communications on Applied Mathematics and Computation, 2019, 1(3): 361-401.
[4]	Jianguo Huang, Xuehai Huang. A Two-Level Additive Schwarz Preconditioner for Local C⁰ Discontinuous Galerkin Methods of Kirchhof Plates[J]. Communications on Applied Mathematics and Computation, 2019, 1(2): 167-185.
[5]	Mark Ainsworth, Zhiping Mao. Analysis and Approximation of Gradient Flows Associated with a Fractional Order Gross-Pitaevskii Free Energy[J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 5-19.
[6]	Jun Hu, Shangyou Zhang. A Cubic H³-Nonconforming Finite Element[J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 81-100.
[7]	Hui Xie, Xuejun Xu. Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems[J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 141-165.