1. Fatih Ecevit Max Planck Institute for Mathematics in the Sciences V í ctor Dom í nguez Ivan Graham New Galerkin Methods for High-frequency Scattering Simulations Universidad Pública de Navarra University of Bath Collaborations

2. Outline High-frequency integral equation methods   Main principles (BGMR 2004)   A robust Galerkin scheme (DGS 2006)   Required improvements II. New Galerkin methods for high-frequency scattering simulations III.   Two new algorithms Electromagnetic & acoustic scattering problems I.I. New Galerkin methods for high-frequency scattering simulations

3. Governing Equations (TE, TM, Acoustic) Maxwell Eqns. Helmholtz Eqn. Electromagnetic & Acoustic Scattering Simulations I.I.

4. Scattering Simulations Basic Challenges: Fields oscillate on the order of wavelength   Computational cost   Memory requirement   Variational methods (MoM, FEM, FVM,…)   Differential Eqn. methods (FDTD,…)   Integral Eqn. methods (FMM, H-matrices,…)   Asymptotic methods (GO, GTD,…) Numerical Methods: Convergent (error-controllable) Demand resolution of wavelength Non-convergent (error ) Discretization independent of frequency Electromagnetic & Acoustic Scattering Simulations I.I.

5. Scattering Simulations Basic Challenges: Fields oscillate on the order of wavelength   Computational cost   Memory requirement   Variational methods (MoM, FEM, FVM,…)   Differential Eqn. methods (FDTD,…)   Integral Eqn. methods (FMM, H-matrices,…)   Asymptotic methods (GO, GTD,…) Numerical Methods: Convergent (error-controllable) Demand resolution of wavelength Non-convergent (error ) Discretization independent of frequency Combine… Electromagnetic & Acoustic Scattering Simulations I.I.

6. Integral Equation Formulations Radiation Condition: High-frequency Integral Equation Methods II. Boundary Condition:

7. Integral Equation Formulations Radiation Condition: Single layer potential: High-frequency Integral Equation Methods II. Boundary Condition: Double layer potential:

8. Integral Equation Formulations Radiation Condition: Single layer potential: High-frequency Integral Equation Methods II.1stkind 2ndkind Boundary Condition: Double layer potential: 2ndkind

9. Single Convex Obstacle: Ansatz Single layer density: High-frequency Integral Equation Methods II. Double layer density:

10. Single Convex Obstacle: Ansatz Single layer density: High-frequency Integral Equation Methods II. Double layer density:

11. Single Convex Obstacle: Ansatz Single layer density: High-frequency Integral Equation Methods II. Double layer density:

12. Single Convex Obstacle: Ansatz Single layer density: High-frequency Integral Equation Methods II. Double layer density: current non-physical is Bruno, Geuzaine, Monro, Reitich (2004)

13. Single Convex Obstacle: Ansatz Single layer density: High-frequency Integral Equation Methods II.

14. Single Convex Obstacle: Ansatz Single layer density: High-frequency Integral Equation Methods II. BGMR (2004)

15. Single Convex Obstacle A Convergent High-frequency Approach Highly oscillatory! High-frequency Integral Equation Methods II.

16. Single Convex Obstacle A Convergent High-frequency Approach Localized Integration: Highly oscillatory! High-frequency Integral Equation Methods II. for all n BGMR (2004)

17. Single Convex Obstacle A Convergent High-frequency Approach High-frequency Integral Equation Methods II. ( Melrose & Taylor, 1985 )

18. Single Convex Obstacle A Convergent High-frequency Approach High-frequency Integral Equation Methods II. ( Melrose & Taylor, 1985 ) Change of Variables: BGMR (2004)

19. Single Smooth Convex Obstacle High-frequency Integral Equation Methods II.   Bruno, Geuzaine, Monro, Reitich … 2004 …   Bruno, Geuzaine (3D) ……………. 2006 …

20. Single Smooth Convex Obstacle High-frequency Integral Equation Methods II.   Bruno, Geuzaine, Monro, Reitich … 2004 …   Bruno, Geuzaine (3D) ……………. 2006 …   Huybrechs, Vandewalle …….…… 2006 …

21. Single Smooth Convex Obstacle High-frequency Integral Equation Methods II.   Domínguez, Graham, Smyshlyaev … 2006 … (circler bd.)   Bruno, Geuzaine, Monro, Reitich … 2004 …   Bruno, Geuzaine (3D) ……………. 2006 …   Huybrechs, Vandewalle …….…… 2006 …

22. Single Smooth Convex Obstacle High-frequency Integral Equation Methods II.   Domínguez, Graham, Smyshlyaev … 2006 … (circler bd.)   Bruno, Geuzaine, Monro, Reitich … 2004 …   Bruno, Geuzaine (3D) ……………. 2006 …   Chandler-Wilde, Langdon ….…….. 2006..   Langdon, Melenk …………..……… 2006.. Single Convex Polygon   Huybrechs, Vandewalle …….…… 2006 …

23. Single Smooth Convex Obstacle High-frequency Integral Equation Methods II.   Domínguez, Graham, Smyshlyaev … 2006 … (circler bd.)   Bruno, Geuzaine, Monro, Reitich … 2004 …   Bruno, Geuzaine (3D) ……………. 2006 …   Chandler-Wilde, Langdon ….…….. 2006..   Langdon, Melenk …………..……… 2006.. Single Convex Polygon   Huybrechs, Vandewalle …….…… 2006 …   Domínguez, E., Graham, ………… 2007 … (circler bd.)

24. The Combined Field Operator A High-frequency Galerkin Method DGS (2006) II.

25. The Combined Field Operator Continuity: circler domains …………… general smooth domains … Giebermann (1997) DGS (2006) II. A High-frequency Galerkin Method DGS (2006)

26. The Combined Field Operator II. Continuity: Coercivity: circler domains …………… general smooth domains … circler domains …………… general smooth domains … open problem Giebermann (1997) DGS (2006) A High-frequency Galerkin Method DGS (2006)

27. Plane-wave Scattering Problem II. A High-frequency Galerkin Method DGS (2006)

28. Plane-wave Scattering Problem II. is an explicitly defined entire function with known asymptotics are smooth periodic functions is not explicitly known but behaves like: A High-frequency Galerkin Method DGS (2006)

29. Plane-wave Scattering Problem II. is an explicitly defined entire function with known asymptotics are smooth periodic functions is not explicitly known but behaves like: DGS (2006) Melrose, Taylor (1985) A High-frequency Galerkin Method DGS (2006)

30. Plane-wave Scattering Problem II. A High-frequency Galerkin Method DGS (2006)

31. Plane-wave Scattering Problem II. for some on the “deep” shadow A High-frequency Galerkin Method DGS (2006)

32. Plane-wave Scattering Problem II. DGS (2006) for some on the “deep” shadow A High-frequency Galerkin Method DGS (2006)

33. Polynomial Approximation II. Illuminated Region Deep Shadow Shadow Boundaries A High-frequency Galerkin Method DGS (2006)

34. Polynomial Approximation II. Illuminated Region Deep Shadow Shadow Boundaries … gluing together A High-frequency Galerkin Method DGS (2006)

35. Polynomial Approximation II. Illuminated Region Deep Shadow Shadow Boundaries … gluing together A High-frequency Galerkin Method DGS (2006)

36. Polynomial Approximation II. Illuminated Region Deep Shadow Shadow Boundaries … gluing together … approximation by zero A High-frequency Galerkin Method DGS (2006)

37. Polynomial Approximation II. Illuminated Region Deep Shadow Shadow Boundaries … gluing together is the optimal choice A High-frequency Galerkin Method DGS (2006)

38. Galerkin Method II. Illuminated Region Deep Shadow Shadow Boundaries … gluing together Discrete space A High-frequency Galerkin Method DGS (2006)

39. Galerkin Method II. Illuminated Region Deep Shadow Shadow Boundaries … gluing together Final Estimate A High-frequency Galerkin Method DGS (2006)

40. Galerkin Method II. Illuminated Region Deep Shadow Shadow Boundaries … gluing together Final Estimate Question Can one obtain a robust Galerkin method that works for higher frequencies as well as low frequencies? A High-frequency Galerkin Method DGS (2006)

41. Galerkin Method II. Illuminated Region Deep Shadow Shadow Boundaries … gluing together Final Estimate A High-frequency Galerkin Method DGS (2006) In other words higher frequencies: low frequencies: do an approximation on the deep shadow region??

42. Galerkin Method II. Illuminated Region Deep Shadow Shadow Boundaries … gluing together Final Estimate A High-frequency Galerkin Method DGS (2006)

43. Galerkin Method II. Illuminated Region Deep Shadow Shadow Boundaries … gluing together Final Estimate A High-frequency Galerkin Method DGS (2006) In other words higher frequencies: low frequencies: do an approximation on the deep shadow region??

44. New Galerkin Methods III. Illuminated Region Deep Shadow Shadow Boundaries … gluing together … new Galerkin methods Treat these four transition regions separately A straightforward extension of the Galerkin approximation in DGS (2006) applies to deep shadow region New Galerkin methods for high-frequency scattering simulations

45. New Galerkin Methods Illuminated Region Deep Shadow Shadow Boundaries … gluing together … new Galerkin methods Treat these four transition regions separately A straightforward extension of the Galerkin approximation in DGS (2006) applies to deep shadow region The highly oscillatory integrals arising in the Galerkin matrices can be efficiently evaluated as the stationary phase points are apriory known III. New Galerkin methods for high-frequency scattering simulations

46. New Galerkin Methods III. New Galerkin methods for high-frequency scattering simulations

47. New Galerkin Methods III. New Galerkin methods for high-frequency scattering simulations

48. New Galerkin Methods III. New Galerkin methods for high-frequency scattering simulations

49. New Galerkin Methods III. New Galerkin methods for high-frequency scattering simulations

50. New Galerkin Methods III. New Galerkin methods for high-frequency scattering simulations

51. New Galerkin Methods III. New Galerkin methods for high-frequency scattering simulations

52. New Galerkin Methods … optimal III. New Galerkin methods for high-frequency scattering simulations

53. New Galerkin Methods Discrete space DGS (2006) III. New Galerkin methods for high-frequency scattering simulations

54. New Galerkin Methods Discrete space DGS (2006) DEG (2007) Discrete space defined in a similar way including the deep shadow … first algorithm III. New Galerkin methods for high-frequency scattering simulations

55. New Galerkin Methods Discrete space DGS (2006) DEG (2007) Discrete space defined in a similar way including the deep shadow … first algorithm III. New Galerkin methods for high-frequency scattering simulations degrees of freedom

56. New Galerkin Methods III. New Galerkin methods for high-frequency scattering simulations

57. New Galerkin Methods III. New Galerkin methods for high-frequency scattering simulations

58. New Galerkin Methods Idea: change of variables III. New Galerkin methods for high-frequency scattering simulations

59. New Galerkin Methods … change of variables III. New Galerkin methods for high-frequency scattering simulations

60. New Galerkin Methods … change of variables control: derivatives of III. New Galerkin methods for high-frequency scattering simulations

61. New Galerkin Methods … change of variables control: derivatives of … but how do we obtain an optimal change of variables? III. New Galerkin methods for high-frequency scattering simulations

62. New Galerkin Methods … change of variables control: derivatives of … but how do we obtain an optimal change of variables? … mimic the algorithm and with affine st. III. New Galerkin methods for high-frequency scattering simulations

63. New Galerkin Methods … change of variables control: derivatives of … but how do we obtain an optimal change of variables? … mimic the algorithm and with affine st. III. New Galerkin methods for high-frequency scattering simulations

64. New Galerkin Methods Discrete space DGS (2006) DEG (2007) … second algorithm Discrete space defined in a similar way including the deep shadow while on the transition regions polynomials are replaced by III. New Galerkin methods for high-frequency scattering simulations

65. New Galerkin Methods Discrete space DGS (2006) DEG (2007) … first algorithm Discrete space defined in a similar way including the deep shadow degrees of freedom DEG (2007) … second algorithm Discrete space defined in a similar way including the deep shadow while on the transition regions polynomials are replaced by III. New Galerkin methods for high-frequency scattering simulations

66. References O. P. Bruno, C. A. Geuzaine, J. A. Monro and F. Reitich: Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: the convex case, Phil. Trans. Roy. Soc. London 362 (2004), 629-645. New Galerkin methods for high-frequency scattering simulations D. Huybrechs and S. Vandewalle: A sparse discretisation for integral equation formulations of high frequency scattering problems, SIAM J. Sci. Comput., (to appear). V. Domínguez, I. G. Graham and V. P. Smyshlyaev: A hybrid numerical-asymptotic boundary integral method for high- frequency acoustic scattering, Num. Math. 106 (2007) 471-510. V. Domínguez, F. Ecevit and I. G. Graham: Improved Galerkin methods for integral equations arising in high- frequency acoustic scattering, (in preparation).

67. Thanks