zbMATH — the first resource for mathematics

Solution of 3D-Laplace and Helmholtz equations in exterior domains using $$hp$$-infinite elements. (English) Zbl 0881.73126
Summary: This work is devoted to a convergence study for infinite element discretizations for Laplace and Helmholtz equations in exterior domains. The proposed approximation applies to separable geometries only, combining an $$hp$$ FE discretization on the boundary of the domain with a spectral-like representation (resulting from the separation of variables) in the ‘radial’ direction. The presentation includes a convergence proof for the Laplace equation and a stability analysis for the variational formulation of the Helmholtz equation in weighted Sobolev spaces. The theoretical investigations are verified and illustrated with numerical examples for the exterior spherical domain.

MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74J20 Wave scattering in solid mechanics 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
GMP
Full Text:
References:
 [1] Astley, R.J.; Macaulay, G.J.; Coyette, J.P., Mapped wave envelope elements for acoustical radiation and scattering, J. sound vib., 170, 1, 97-118, (1994) · Zbl 0925.76419 [2] Cremers, L.; Fyfe, K.R.; Coyette, J.P., A variable order infinite acoustic wave envelope element, J. sound vib., 171, 4, 483-508, (1994) · Zbl 1059.76516 [3] Cremers, L.; Fyfe, K.R., On the use of variable order infinite wave envelope elements for acoustic radiation and scattering, J. acoustic. am., 97, 4, 2028-2040, (1995) [4] Bettess, P.; Bettess, J.A., Infinite elements for dynamic problems: part 1, Engrg. comput., 8, 99-124, (1991) [5] Bettess, P.; Bettess, J.A., Infinite elements for dynamic problems: part 2, Engrg. comput., 8, 125-151, (1991) [6] Burnett, D.S., A three-dimensional acoustic infinite element based on a prolate spheroidal multipole expansion, J. acoustic. am., 96, 2798-2816, (1994) [7] Demkowicz, L.; Oden, J.T., Application of HP-adaptive BE/FE methods to elastic scattering, TICAM report 94-15, (December 1994) [8] Demkowicz, L., Asymptotic convergence in finite and boundary element methods: part 1: theoretical results, Comput. math. appl., 27, 12, 69-84, (1994) · Zbl 0807.65058 [9] Demkowicz, L., Asymptotic convergence in finite and boundary element methods: part 2: the LBB constant for rigid and elastic scattering problems, Comput. math. appl., 28, 6, 93-109, (1994) · Zbl 0818.73071 [10] Demkowicz, L.; Bajer, A.; Banas, K., Geometrical modeling package, () [11] Demkowicz, L.; Karafiat, A.; Oden, J.T., Solution of elastic scattering problems in linear acoustics using hp boundary element method, Comput. methods appl. mech. engrg., 101, 251-282, (1992) · Zbl 0778.73081 [12] Geng, P.; Oden, J.T.; van de Geijn, R.A., Massively parallel computation for acoustical scattering problems using boundary element methods, TICAM report 94-10, (August 1994) [13] Ihlenburg, F.; Babuska, I., Finite element solution to the Helmholtz equation with high wave number: part 1: the h-version of the FEM, Comput. math. applic., 30, 9, 9-37, (1995) · Zbl 0838.65108 [14] Leis, R., Initial boundary value problems in mathematical physics, (1986), Teubner · Zbl 0599.35001 [15] Morse, P.M.; Feshbach, J., () [16] Nedelec, J.C., Curved finite element methods for the solution of singular integral equations on surfaces in $$R$$^{3}, Comput. methods appl. mech. engrg., 8, 61-80, (1976) · Zbl 0333.45015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.