Goldstein, C. I. (1982): A Finite Element Method for Solving Helmholtz Type Equations in Waveguides and Other Unbounded Domains, Math. Comp., 39, pp. 309-324  Home/Search   Document Not in Database   Summary   Related Articles   Check

....deal with a general elliptic boundary value problem of the form ) b(x)u(x) f(x) on an unbounded domain # which is an exterior of a bounded domain. Over the years, related to the Finite Element Method (FEM) 9] 29] much work has been done on unbounded domain problems ( 1] 9] 11] [12], 14] 15] 22] 28] Several numerical methods for these problems were suggested. The following approaches are the most typical: truncating the unbounded part of the domain and introducing an artificial boundary condition on the resulting artificial boundary; coupling boundary element method ....

Goldstein, C. I. (1982): A Finite Element Method for Solving Helmholtz Type Equations in Waveguides and Other Unbounded Domains, Math. Comp., 39, pp. 309-324

....of corresponding local and asymptotic approximations [6] infinite element methods [7] and the various kinds of boundary element methods [8] However, if the exterior domain becomes inhomogeneous, e.g. by an embedded waveguide (Fig. 1) these methods do not longer work. For such cases Goldstein [9] offers a solution by constructing transparent boundary conditions valid on a part of the whole boundary. This part of the boundary contains the position where the waveguide hits the boundary and the area surrounding the waveguide. In Fig. 1 this would be the left, vertical part of the boundary. ....

C. I. Goldstein. A finite-element method for solving Helmholtz type equations in waveguides and other unbounded domains. Math. of Comput. , 39, 1982.

....a tool like the Sommerfeld condition at hand, in order to decide, what is outgoing and what is incoming. However, in some practical relevant situations such a tool is not available, for example in cases with non constant coefficients or even in some cases with piecewise constant coefficients. In [5] Goldstein investigated the case of an waveguide type inhomogeneity. In examples like this the Sommerfeld condition is not applicable. The purpose of this paper is to give a new view on Sommerfeld s radiation condition. We do not aim to give a new boundary conditions or a new numerical method but ....

C. I. Goldstein. A finite-element method for solving Helmholtz type equations in waveguides and other unbounded domains. Math. of Comput., 39(160):309--324, 1982.

.... elliptic boundary value problem of the form Gamma n X i;j=1 x j (a ij (x) u(x) x i ) b(x)u(x) f(x) on an unbounded domain Omega Gamma Over the years, related to the Finite Element Method (FEM) 9] 29] much work has been done on unbounded domain problems ( 1] 9] 11] [12], 15] 16] 23] 28] and several numerical methods for these problems were suggested. The following approaches are the most typical: truncating the unbounded part of the domain and introducing an artificial boundary condition on the resulting artificial boundary; coupling boundary element ....

Goldstein, C. I. (1982): A Finite Element Method for Solving Helmholtz Type Equations in Waveguides and Other Unbounded Domains, Math. Comp., 39, pp. 309-324

....a general elliptic boundary value problem of the form Gamma n X i;j=1 x j (a ij (x) u(x) x i ) b(x)u(x) f(x) on a unbounded domain Omega . Over the years, related to the Finite Element Method (FEM) 9] 30] much work has been done on unbounded domain problems ( 1] 9] 11] [12], 15] 16] 23] 29] and several numerical methods for these problems were suggested. The following approaches are the most typical: truncating the unbounded part of the domain, coupling boundary element method with FEM ( 20] and using infinite elements ( 6] 7] 8] 32] The basic idea ....

Goldstein, C. I. (1982): A Finite Element Method for Solving Helmholtz Type Equations in Waveguides and Other Unbounded Domains, Math. Comp., 39, pp. 309-324

....domain, it must be supplied with the Sommerfeld radiation condition at infinity [71] which guarantees that waves are purely outgoing and decaying as they approach infinity. A general discussion on finite element methods for solving the reduced wave equation in unbounded domains can be found in [22]. Up to now, boundary integral or boundary element methods based on integral formulations have been the methods of choice for the time harmonic problem. For a historical survey of boundary integral methods see Shaw [69] A comprehensive review can also be found in Givoli [16] The boundary ....

C. I. Goldstein. A finite-element method for solving Helmholtz type equations in waveguides and other unbounded domains. Math. of Comput., 39(160):309--324, 1982.

....on Gamma. Recently, Kirsch and Monk [72] have used the integral operator representation of the DtN map to obtain the DtN boundary condition for arbitrary smooth artificial boundaries using a Nystrom boundary integral discretization. Other applications of the DtN idea include waveguide problems [54] and underwater acoustics [39] Circular Artificial Boundaries When the artificial boundary is a circle, the DtN mapping for the Helmholtz equation can be obtained by using a series expansion of the solution obtained by separation of variables. The solution of the exterior Helmholtz problem ....

....d (1.28) M k u(R; Delta) #) 1. 29) where the DtN mapping M k is seen to be a (pseudo differential) operator on the space of functions defined on Gamma R or, equivalently, on [0; 2 ] Other references which use a separation of variables representation for the DtN mapping include [39, 54, 80]. Exactness of the DtN Boundary Condition The exactness of the DtN boundary condition, i.e. the fact that imposing it on the artificial boundary of a truncated exterior domain results in a solution which is the CHAPTER 1. INTRODUCTION 34 restriction of the solution of the exterior problem to ....

C.I. Goldstein. A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains. Math. Comp., 39(160):309--324, October 1982.

....third problem, a full vector model generally needs to be retained, although the geometry is still two dimensional . All three cases, as well as the case of the full Maxwell equations in three dimensions have been studied extensively in the engineering and mathematical literature; see for example [1, 2, 6, 11, 13, 17, 20, 25, 27, 32] and the references therein. For simplicity, here we only consider the first and simplest case, TE polarization. Let E = u x 3 where u = u(x 1 ; x 2 ) is a scalar function. The Maxwell equations then reduce to the Helmholtz equation: 4 k 2 )u = 0 ; in IR 2 ; 4) where k is the index of ....

C. Goldstein, A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains, Math. Comp., 39 (1982), 309-324.

....Due to the ill conditioning of the normal equations, the unpreconditioned algorithm suffered from extremely slow convergence. The convergence rate was substantially improved through preconditioners based on symmetric successive overrelaxation [BaGoTu83] or a multigrid V cycle [BaGoTu85b] [Gold82]; only for the Laplacian part of the Helmholtz operator. Recently, the iterative quasi minimal residual algorithm has been applied to capacitance matrix methods for exterior Helmholtz problems [Ernst94] The objective of this paper is to develop a technique for solving the Helmholtz equation with ....

C. I. Goldstein, A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains, Math. Comp., 39 (1982), pp. 309-- 324.

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