Fast parallel solvers for symmetric boundary element domain decomposition equations

Summary. The boundary element method (BEM) is of advantage in many applications including far-field computations in magnetostatics and solid mechanics as well as accurate computations of singularities. Since the numerical approximation is essentially reduced to the boundary of the domain under consideration, the mesh generation and handling is simpler than, for example, in a finite element discretization of the domain. In this paper, we discuss fast solution techniques for the linear systems of equations obtained by the BEM (BE-equations) utilizing the non-overlapping domain decomposition (DD). We study parallel algorithms for solving large scale Galerkin BE–equations approximating linear potential problems in plane, bounded domains with piecewise homogeneous material properties. We give an elementary spectral equivalence analysis of the BEM Schur complement that provides the tool for constructing and analysing appropriate preconditioners. Finally, we present numerical results obtained on a massively parallel machine using up to 128 processors, and we sketch further applications to elasticity problems and to the coupling of the finite element method (FEM) with the boundary element method. As shown theoretically and confirmed by the numerical experiments, the methods are of algebraic complexity and of high parallel efficiency, where denotes the usual discretization parameter. Received August 28, 1996 / Revised version received March 10, 1997     

Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations

Numerical Algorithms - Generally, solving linear systems from finite difference alternating direction implicit scheme of two-dimensional time-space fractional differential equations with Gaussian...     

Fast direct solvers for Poisson equation on 2D polar and spherical geometries

A simple and efficient class of FFT‐based fast direct solvers for Poisson equation on 2D polar and spherical geometries is presented. These solvers rely on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by the second‐ and fourth‐order finite difference discretizations. Using a grid by shifting half mesh away from the origin/poles, and incorporating with the symmetry constraint of Fourier coefficients, the coordinate singularities can be easily handled without pole condition. By manipulating the radial mesh width, three different boundary conditions for polar geometry including Dirichlet, Neumann, and Robin conditions can be treated equally well. The new method only needs O(MN log₂ N) arithmetic operations for M × N grid points. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 56–68, 2002     

Fast direct solver for Poisson equation in a 2D elliptical domain

In this article, we extend our previous work 3 for developing some fast Poisson solvers on 2D polar and spherical geometries to an elliptical domain. Instead of solving the equation in an irregular Cartesian geometry, we formulate the equation in elliptical coordinates. The solver relies on representing the solution as a truncated Fourier series, then solving the differential equations of Fourier coefficients by finite difference discretizations. Using a grid by shifting half mesh away from the pole and incorporating the derived numerical boundary value, the difficulty of coordinate singularity can be elevated easily. Unlike the case of 2D disk domain, the present difference equation for each Fourier mode is coupled with its conjugate mode through the numerical boundary value near the pole; thus, those two modes are solved simultaneously. Both second‐ and fourth‐order accurate schemes for Dirichlet and Neumann problems are presented. In particular, the fourth‐order accuracy can be achieved by a three‐point compact stencil which is in contrast to a five‐point long stencil for the disk case. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 72–81, 2004     

Solvability of a finite or infinite system of discontinuous quasimonotone differential equations

This paper proves the existence of solutions to the initial value problem

where may be discontinuous but is assumed to satisfy conditions of superposition-measurability, quasimonotonicity, quasisemicontinuity, and integrability. The set can be arbitrarily large (finite or infinite); our theorem is new even for . The proof is based partly on measure-theoretic techniques used in one dimension under slightly stronger hypotheses by Rzymowski and Walachowski. Further generalizations are mentioned at the end of the paper.

     

Two regularity criteria for 3D Navier-Stokes equations in a bounded domain

We prove two new regularity criteria for the 3D incompressible Navier-Stokes equations in a bounded domain. Our results also hold for the 3D Boussinesq system with zero heat conductivity.     

On domain decomposition preconditioner of BPS type for finite element discretizations of 3D elliptic equations

BPS is a well known an efficient and rather general domain decomposition Dirichlet-Dirichlet type preconditioner, suggested in the famous series of papers Bramble, Pasciak and Schatz (1986?C1989). Since then, it has been serving as the origin for the whole family of domain decomposition Dirichlet-Dirichlet type preconditioners-solvers as for h so hp discretizations of elliptic problems. For its original version, designed for h discretizations, the named authors proved the bound O(1 + log² H/h) for the relative condition number under some restricting conditions on the domain decomposition and finite element discretization. Here H/h is the maximal relation of the characteristic size H of a decomposition subdomain to the mesh parameter h of its discretization. It was assumed that subdomains are images of the reference unite cube by trilinear mappings. Later similar bounds related to h discretizations were proved for more general domain decompositions, defined by means of coarse tetrahedral meshes. These results, accompanied by the development of some special tools of analysis aimed at such type of decompositions, were summarized in the book of Toselli and Widlund (2005). This paper is also confined to h discretizations. We further expand the range of admissible domain decompositions for constructing BPS preconditioners, in which decomposition subdomains can be convex polyhedrons, satisfying some conditions of shape regularity. We prove the bound for the relative condition number with the same dependence on H/h as in the bound given above. Along the way to this result, we simplify the proof of the so called abstract bound for the relative condition number of the domain decomposition preconditioner. In the part, related to the analysis of the interface sub-problem preconditioning, our technical tools are generalization of those used by Bramble, Pasciak and Schatz.     

High-order compact solvers for the three-dimensional Poisson equation

New compact approximation schemes for the Laplace operator of fourth- and sixth-order are proposed. The schemes are based on a Padé approximation of the Taylor expansion for the discretized Laplace operator. The new schemes are compared with other finite difference approximations in several benchmark problems. It is found that the new schemes exhibit a very good performance and are highly accurate. Especially on large grids they outperform noncompact schemes.     

Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method

In this paper, we present a computational method for solving 2D and 3D Poisson equations and biharmonic equations which based on the use of Haar wavelets. The highest derivative appearing in the differential equation is expanded into the Haar series, this approximation is integrated while the boundary conditions are incorporated by using integration constants. In 2D the first transform the spectral coefficients into the nodal variable values and then use Kronecker products to construct the approximations for derivatives over a tensor product grid of the horizontal and vertical blocks. Finally, solutions to four test problems are investigated.     

Global well-posedness and time-decay for the full compressible non-resistive MHD equations in a 3D infinite slab

This paper is concerned with the global well-posedness and time-decay of the system of full compressible viscous non-resistive MHD fluids in a three-dimensional horizontally infinite slab with finite height. We reformulate our analysis to Lagrangian coordinates, and then develop a new mathematical approach to establish global well-posedness of the MHD system, which requires no compatibility conditions on the initial data.     

On some boundary value problems for a system of Poisson equations in a three-dimensional domain

For a system of Poisson equations in a three-dimensional domain, we consider two nonstandard problems with vector boundary conditions. On the basis of inequalities of the Friedrichs type, we show that these problems are well posed in the Hadamard-Petrovskii sense.     

Efficient iterative solvers for time-harmonic Maxwell equations using domain decomposition and algebraic multigrid

Paper presents a set of parallel iterative solvers and preconditioners for the efficient solution of systems of linear equations arising in the high order finite-element approximations of boundary value problems for 3-D time-harmonic Maxwell equations on unstructured tetrahedral grids. Balancing geometric domain decomposition techniques combined with algebraic multigrid approach and coarse-grid correction using hierarchic basis functions are exploited to achieve high performance of the solvers and small memory load on the supercomputers with shared and distributed memory. Testing results for model and real-life problems show the efficiency and scalability of the presented algorithms.     

The 3D compressible Euler equations with damping in a bounded domain

We proved global existence and uniqueness of classical solutions to the initial boundary value problem for the 3D damped compressible Euler equations on bounded domain with slip boundary condition when the initial data is near its equilibrium. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on energy estimate, we showed that the classical solution converges to steady state exponentially fast in time. We also proved that the same is true for the related initial boundary value problem of porous medium equation and thus justified the validity of Darcy's law in large time.     

On the global exponential stability of impulsive functional differential equations with infinite delays or finite delays

This paper considers the impulsive functional differential equations with infinite delays or finite delays. Some new sufficient conditions are obtained to guarantee the global exponential stability by employing the improved Razumikhin technique and Lyapunov functions. The result extends and improves some recent works. Moreover, the obtained Razumikhin condition is very simple and effective to implement in real problems and it is helpful to investigate the stability of delayed neural networks and synchronization problems of chaotic systems under impulsive perturbation. Finally, a numerical example and its simulation is given to show the effectiveness of the obtained result in this paper.     

Global bifurcation for equations involving nonhomogeneous operators in an unbounded domain

We study the structure of the set of solutions of a nonlinear equation involving nonhomogeneous operators:

     

Haar wavelets method for solving Poisson equations with jump conditions in irregular domain

In this paper, Haar wavelets method is used to solve Poisson equations in the presence of interfaces where the solution itself may be discontinuous. The interfaces have jump conditions which need to be enforced. It is critical for the approximation of the boundaries of the irregular domain. An irregular domain can be treated by embedding the domain into a rectangular domain and Poisson equation is solved by using Haar wavelets method on the rectangle. Firstly, we demonstrate this method in the case of 1-D region, then we consider the solution of the Poisson equations in the case of 2-D region. The efficiency of the method is demonstrated by some numerical examples.