Microwave tomography via domain decomposition for finite element methods

Abstract Based upon full domain decomposition, local and parallel stabilized finite element methods for the stationary Stokes equations are proposed and analysed, where the quadratic equal-order finite elements are employed for the velocity and pressure approximations, and a stabilized term based on two local Gauss integrations is used to offset the discrete pressure space to circumvent the discrete inf-sup condition. In the proposed parallel method, all of the computations are performed on the locally refined global grids that are fine around the interested subdomain and coarse elsewhere, making the method easy to implement based on a sequential solver with low communication cost. Stability and optimal error estimates of the present methods are deduced. Numerical results on examples including a problem with known analytic solution, lid-driven cavity flow, backward-facing step flow and flow around a cylinder are given to verify the theoretical predictions and demonstrate the high efficiency of the method. Results show that our parallel method can provide an approximate solution with the convergence rate of the same order as the solution computed by the standard stabilized finite element method, with a substantial reduction in computational time.

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Adaptive Finite Element Methods for Domain Decomposition on Nonmatching Grids

Parallel Solution of Partial Differential Equations - The IMA Volumes in Mathematics and its Applications ◽

10.1007/978-1-4612-1176-1_3 ◽

2000 ◽

pp. 57-83 ◽

Cited By ~ 1

Author(s):

B. Engelmann ◽

R. H. W. Hoppe ◽

Y. Iliash ◽

Y. A. Kuznetsov ◽

Y. Vassilevski ◽

...

Keyword(s):

Finite Element ◽

Domain Decomposition ◽

Finite Element Methods ◽

Adaptive Finite Element ◽

Adaptive Finite Element Methods ◽

Nonmatching Grids

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NONCONFORMING FINITE ELEMENT METHODS FOR THE THREE-DIMENSIONAL HELMHOLTZ EQUATION: ITERATIVE DOMAIN DECOMPOSITION OR GLOBAL SOLUTION?

Journal of Computational Acoustics ◽

10.1142/s0218396x09003902 ◽

2009 ◽

Vol 17 (02) ◽

pp. 159-173 ◽

Cited By ~ 6

Author(s):

PATRICIA M. GAUZELLINO ◽

FABIO I. ZYSERMAN ◽

JUAN E. SANTOS

Keyword(s):

Finite Element ◽

Helmholtz Equation ◽

Domain Decomposition ◽

Finite Element Methods ◽

Three Dimensional ◽

Domain Size ◽

Nonconforming Finite Element ◽

Parallel Performance ◽

Nonconforming Finite Element Methods ◽

Better Than

Iterative domain decomposition (DD) nonconforming finite element methods for the Helmholtz equation attempt to solve two problems. First, there exists no efficient algorithms able to solve the large sparse linear system arising from the discretization of the equation via the standard finite elements method. Secondly, even when DD methods generally yield small matrices, standard conforming elements, such as Q1 elements, force the transmission of a relatively large amount of data among subdomains. In this paper, we compared performance of global methods and a set of DD techniques to solve the Helmholtz equation in a three-dimensional domain. The efficiency of the algorithms is measured in terms of CPU time usage and memory requirements. The role of domain size and the linear solver type used to solve each local problem within each subdomain was also analyzed. The numerical results show that iterative DD methods perform far better than global methods. In addition, iterative DD methods involving small subdomains work better than those with subdomains involving a large number of elements. Properties of the iterative DD algorithms such as scalability, robustness, and parallel performance are also analyzed.

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