Automatic Discretization of a Three-Dimensional Domain into Voronoi Polyhedron Elements

The stationary monochromatic radiative transfer equation (RTE) is posed in five dimensions, with the intensity depending on both a position in a three-dimensional domain as well as a direction. For non-scattering radiative transfer, sparse finite elements [1, 2] have been shown to be an efficient discretization strategy if the intensity function is sufficiently smooth. Compared to the discrete ordinates method, they make it possible to significantly reduce the number of degrees of freedom N in the discretization with almost no loss of accuracy. However, using a direct solver to solve the resulting linear system requires O(N3) operations. In this paper, an efficient solver based on the conjugate gradient method (CG) with a subspace correction preconditioner is presented. Numerical experiments show that the linear system can be solved at computational costs that are nearly proportional to the number of degrees of freedom N in the discretization.

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Alternative Approach to Infinity

Symposium - International Astronomical Union ◽

10.1017/s0074180900236139 ◽

1974 ◽

Vol 64 ◽

pp. 99-99

Author(s):

Peter G. Bergmann

Keyword(s):

Large Class ◽

Riemannian Manifolds ◽

Quotient Space ◽

Three Dimensional ◽

Cauchy Data ◽

Space Time ◽

Equivalence Classes ◽

Null Infinity ◽

Alternative Approach ◽

Dimensional Domain

Following Penrose's construction of space-time infinity by means of a conformal construction, in which null-infinity is a three-dimensional domain, whereas time- and space-infinities are points, Geroch has recently endowed space-infinity with a somewhat richer structure. An approach that might work with a large class of pseudo-Riemannian manifolds is to induce a topology on the set of all geodesics (whether complete or incomplete) by subjecting their Cauchy data to (small) displacements in space-time and Lorentz rotations, and to group the geodesics all of whose neighborhoods intersect into equivalence classes. The quotient space of geodesics over equivalence classes is to represent infinity. In the case of Minkowski, null-infinity has the usual structure, but I0, I+, and I- each become three-dimensional as well.

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The Unfolding Story of Three-Dimensional Domain Swapping

Structure ◽

10.1016/s0969-2126(03)00029-7 ◽

2003 ◽

Vol 11 (3) ◽

pp. 243-251 ◽

Cited By ~ 164

Author(s):

Frederic Rousseau ◽

Joost W.H. Schymkowitz ◽

Laura S. Itzhaki

Keyword(s):

Three Dimensional ◽

Domain Swapping ◽

Dimensional Domain

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A D-N Alternating Algorithm for Solving 3D Exterior Helmholtz Problems

Mathematical Problems in Engineering ◽

10.1155/2014/418426 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Qing Chen ◽

Baoqing Liu ◽

Qikui Du

Keyword(s):

Domain Decomposition Method ◽

Three Dimensional ◽

Boundary Element Methods ◽

The Finite Element Method ◽

Natural Boundary ◽

Artificial Boundary ◽

Nonoverlapping Domain ◽

Helmholtz Problem ◽

Nonoverlapping Domain Decomposition ◽

Dimensional Domain

The nonoverlapping domain decomposition method, which is based on the natural boundary reduction, is applied to solve the exterior Helmholtz problem over a three-dimensional domain. The basic idea is to introduce a spherical artificial boundary; the original unbounded domain is changed into a bounded subdomain and a typical unbounded region; then, a Dirichlet-Nuemann (D-N) alternating method is presented; the finite element method and natural boundary element methods are alternately applied to solve the problems in the bounded subdomain and the typical unbounded subdomain. The convergence of the D-N alternating algorithm and its discretization are studied. Some numerical experiments are presented to show the performance of this method.

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