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Fictitious domain methods for the numerical solution of two-dimensional scattering problems. (English) Zbl 0909.65119
The authors consider the two-dimensional exterior Helmholtz equation \[ \begin{gathered} \triangle u+\omega^2u=0\;\text{in} \mathbb{R}^2\backslash\overline\Omega,\quad u=-v_I\;\text{on} \partial\Omega\\ \lim_{r\to\infty}\sqrt{r}\left(\frac{\partial u}{\partial r}-{\mathbf i}\omega u\right)=0, \end{gathered} \] which can be used to model the scattering of time-harmonic electromagnetic or acoustic waves by an obstacle denoted by \(\Omega\). Here, \(v_I({\mathbf x})=e^{i\mathbf{\omega}\cdot{\mathbf x}}\) is the time-harmonic incident plane wave the direction of propagation of which is given by the vector \({\mathbf \omega}\). This exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary. First-order, second-order, and exact nonreflecting boundary conditions are tested on rectangular and circular boundaries. The finite element discretizations of the corresponding approximate boundary value problems are performed using locally fitted meshes, and the discrete equations are solved with fictious domain methods. A special finite element method uses the macro-hybrid formulation based on domain decomposition to couple polar and cartesian coordinate systems. A special preconditioner based on fictitious domains is introduced for the arising algebraic saddle-point system such that the subspace of constraints becomes invariant with respect to the preconditioned iterative procedure. The performance of the new method is compared to the fictitious domain methods both with respect to accuracy and computational cost.
Reviewer: K.Najzar (Praha)

MSC:
65Z05 Applications to the sciences
78A45 Diffraction, scattering
76Q05 Hydro- and aero-acoustics
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q35 PDEs in connection with fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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References:
[1] Jäger, W., Zur theorie der schwingungsgleichung mit variablen koeffizienten in aussengebieten, Math. Z., 102, (1967) · Zbl 0162.16402
[2] Rokhlin, V., Rapid solution of integral equations of scattering theory in two dimensions, J. comput. phys., 86, 2, (1990) · Zbl 0686.65079
[3] Marchuk, G.I.; Kuznetsov, Yu.A.; Matsokin, A.M., Fictitious domain and domain decomposition methods, Soviet J. numer. anal. math. modelling, 1, 3, (1986) · Zbl 0825.65027
[4] Rossi, T., Fictitious domain methods with separable preconditioners, (1995) · Zbl 0835.65056
[5] Bespalov, A., Application of fictitious domain method to the solution of the Helmholtz equation in unbounded domain, (1992)
[6] Bespalov, A.; Kuznetsov, Y.; Pironneau, O.; Vallet, M.-G., Fictitious domains with separable preconditioners versus unstructured adapted meshes, IMPACT comp. sci. eng., 4, (1992) · Zbl 0760.76068
[7] Kuznetsov, Yu.A., Matrix computational processes in subspaces, Comput. math. in appl. sci. and eng. VI, 15, (1984) · Zbl 0564.65014
[8] Yu. A. Kuznetsov, K. Lipnikov, A fictitious domain method for solving the Helmholtz wave equation in unbounded domains, Numer. Meth. Math. Modelling, Yu. A. Kuznetsov, Institute of Numerical Mathematics of Russian Academy of Sciences, Moscow, 1992, 56
[9] Kuznetsov, Yu.A.; Lipnikov, K., On the application of fictitious domain and domain decomposition methods for scattering problems on cray Y-MP C98, (1995)
[10] Yu. A. Kuznetsov, A. M. Matsokin, On partial solution of systems of linear algebraic equations, Vychislitel’nye Metody Lineinoy Algebry (Computational Methods of Linear Algebra), G. I. Marchuk, Vychisl. Tsentr Sib. Otdel. Akad. Nauk SSSR, Novosibirsk, 1978, 62
[11] Heikkola, E.; Kuznetsov, Yu.A., Finite element method on nonmatching meshes for the Helmholtz equation, (1995)
[12] Heikkola, E., Domain decomposition method with nonmatching grids for acoustic scattering problems, (1997) · Zbl 0885.65131
[13] Ernst, O.G., A finite-element capacitance matrix method for exterior Helmholtz problems, Numer. math., 75, (1996) · Zbl 0874.65084
[14] Goldstein, C.I., The finite element method with non-uniform mesh sizes applied to the exterior Helmholtz problem, Numer. math., 38, 61, (1981) · Zbl 0445.65102
[15] Goldstein, C.I., The solution of exterior interface problems using a variational method with Lagrange multipliers, J. math. anal. appl., 97, 480, (1983) · Zbl 0581.65076
[16] Cooray, F.R.; Costache, G.I., An overview of the absorbing boundary conditions, J. electromagn. waves appl., 5, 1041, (1991)
[17] Givoli, D., Non-reflecting boundary conditions, J. comput. phys., 94, 1, (1991) · Zbl 0731.65109
[18] Bayliss, A.; Gunzburger, M.; Turkel, E., Boundary conditions for the numerical solution of elliptic equations in exterior regions, SIAM J. appl. math., 42, 430, (1982) · Zbl 0479.65056
[19] Engquist, B.; Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Math. comp., 31, 629, (1977) · Zbl 0367.65051
[20] Engquist, B.; Majda, A., Radiation boundary conditions for acoustic and elastic wave calculations, Comm. pure appl. math., 32, 313, (1979) · Zbl 0387.76070
[21] Halpern, L.; Rauch, J., Error analysis for the absorbing boundary conditions, Numer. math., 51, 459, (1987) · Zbl 0656.35076
[22] Bamberger, A.; Joly, P.; Roberts, J.E., Second-order absorbing boundary conditions for the wave equation: a solution for the corner problem, SIAM J. numer. anal., 27, 323, (1990) · Zbl 0716.35036
[23] MacCamy, R.C.; Marin, S.P., A finite element method for exterior interface problems, Int. J. math. math. sci., 3, 2, (1980)
[24] Keller, J.B.; Givoli, D., Exact non-reflecting boundary conditions, J. comput. phys., 82, 172, (1989) · Zbl 0671.65094
[25] Harari, I.; Hughes, T.J.R., Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains, Comput. methods appl. mech. eng., 97, 103, (1992) · Zbl 0769.76063
[26] Ihlenburg, F.; Babuška, I., Finite element solution to the Helmholtz equation with high wave number. part I. the h – version of the FEM, Comput. math. appl., 30, 9, (1995) · Zbl 0838.65108
[27] Banegas, A., Fast Poisson solvers for problems with sparsity, Math. comp., 32, 441, (1978) · Zbl 0375.65046
[28] McLachlan, N.W., Theory and application of Mathieu functions, (1964) · Zbl 0128.29603
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