A new family of linearly implicit fractional step methods is proposed for the efficient numerical solution of a class of nonlinear time-dependent reaction-diffusion equations. By using the method of lines, the origina...
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ISBN:
(纸本)9783642026775;9783642026768
A new family of linearly implicit fractional step methods is proposed for the efficient numerical solution of a class of nonlinear time-dependent reaction-diffusion equations. By using the method of lines, the original problem is first discretized in space via a mimetic finite difference technique. The resulting differential system of stiff nonlinear equations is locally decomposed by suitable Taylor expansions and a domain decomposition splitting for the linear terms. This splitting is then combined with a linearly implicit one-step scheme belonging to the class of so-called fractional step Runge-Kutta methods. In this way, the original problem is reduced to the solution of several linear systems per time step which can be trivially decomposed into a set of uncoupled subsystems. As compared to classical domain decomposition techniques, our proposal does not require any Schwarz iterative procedure. The convergence of the designed method is illustrated by numerical experiments.
An iterative substructuring method with Lagrange multipliers is considered for the second order elliptic problem, which is a variant of the FETI-DP method. The standard FETI-DP formulation is associated with a saddle-...
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ISBN:
(纸本)9783642026775;9783642026768
An iterative substructuring method with Lagrange multipliers is considered for the second order elliptic problem, which is a variant of the FETI-DP method. The standard FETI-DP formulation is associated with a saddle-point problem which is induced from the minimization problem with a constraint for imposing the continuity across the interface. Starting from the slightly changed saddle-point problem by addition of a penalty term with a positive penalization parameter eta, we propose a dual substructuring method which is implemented iteratively by the conjugate gradient method. In spite of the absence of any preconditioners, it is shown that the proposed method is numerically scalable in the sense that for a large value of eta, the condition number of the resultant dual problem is bounded by a constant independent of both the subdomain size H and the mesh size h. We discuss computational issues and present numerical results.
Simulations of saturated-unsaturated groundwater flow in heterogeneous soil can be carried out by considering non-overlapping domain decomposition problems for the Richards equation in subdomains with homogeneous soil...
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ISBN:
(纸本)9783642026775;9783642026768
Simulations of saturated-unsaturated groundwater flow in heterogeneous soil can be carried out by considering non-overlapping domain decomposition problems for the Richards equation in subdomains with homogeneous soil. By the application of different Kirchhoff transformations in the different subdomains local convex minimization problems can be obtained which are coupled via superposition operators on the interface between the subdomains. The purpose of this article is to provide a rigorous mathematical foundation for this reformulation in a weak sense. In particular, this involves an analysis of the Kirchhoff transformation as a superposition operator on Sobolev and trace spaces.
Optimized Schwarz methods have been developed at the continuous level;in order to obtain optimized transmission conditions, the underlying partial differential equation (PDE) needs to be known. Classical Schwarz metho...
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ISBN:
(纸本)9783642026775;9783642026768
Optimized Schwarz methods have been developed at the continuous level;in order to obtain optimized transmission conditions, the underlying partial differential equation (PDE) needs to be known. Classical Schwarz methods on the other hand can be used in purely algebraic form, which have made them popular. Their performance can however be inferior compared to that of optimized Schwarz methods. We present in this paper a discovery algorithm, which, based purely on algebraic information, allows us to obtain an optimized Schwarz preconditioner for a large class of numerically discretized elliptic PDEs. The algorithm detects the nature of the elliptic PDE, and then modifies a classical algebraic Schwarz preconditioner at the algebraic level, using existing optimization results from the literature on optimized Schwarz methods. Numerical experiments using elliptic problems discretized by Q(1)-FEM, P-1-FEM, and FDM demonstrate the algebraic nature and the effectiveness of the discovery algorithm.
In this paper, we propose and study a Robin domain decomposition algorithm to approximate a frictionless unilateral problem between two elastic bodies. Indeed this algorithm combines, in the contact zone, the Dirichle...
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ISBN:
(纸本)9783642026775;9783642026768
In this paper, we propose and study a Robin domain decomposition algorithm to approximate a frictionless unilateral problem between two elastic bodies. Indeed this algorithm combines, in the contact zone, the Dirichlet and Neumann boundaries conditions (Robin boundary condition). The primary feature of this algorithm is the resolution on each sub-domain of variational inequality.
We present a new inexact nonsmooth Newton method for the solution of convex minimization problems with piecewise smooth, pointwise nonlinearities. The algorithm consists of a nonlinear smoothing step on the fine level...
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ISBN:
(纸本)9783642026775
We present a new inexact nonsmooth Newton method for the solution of convex minimization problems with piecewise smooth, pointwise nonlinearities. The algorithm consists of a nonlinear smoothing step on the fine level and a linear coarse correction. Suitable postprocessing guarantees global convergence even in the case of a single multigrid step for each linear subproblem. Numerical examples show that the overall efficiency is comparable to multigrid for similar linear problems.
We aim in this paper to give a unified presentation to some important approaches in multi-phase flow in porous media within the framework of multiscale methods. Thereafter, we will present a modern outlook indicating ...
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ISBN:
(纸本)9783642026775;9783642026768
We aim in this paper to give a unified presentation to some important approaches in multi-phase flow in porous media within the framework of multiscale methods. Thereafter, we will present a modern outlook indicating future research directions in this field.
We investigate Schwarz' domain decomposition algorithm as a tool for numerical zoom and compare it with the Subspace Correction Method. Quadrature error is investigated and the convergence of Schwarz' algorith...
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ISBN:
(纸本)9783642026775;9783642026768
We investigate Schwarz' domain decomposition algorithm as a tool for numerical zoom and compare it with the Subspace Correction Method. Quadrature error is investigated and the convergence of Schwarz' algorithm is sketched for non matching grids. The methods are also compared numerically.
These are the proceedings of the 18th international conference on domain decomposition methods in science and engineering, heldin Jerusalem, January 12-17, 2008. Domain decomposition methods are iterative methods for ...
ISBN:
(数字)9783642026775
ISBN:
(纸本)9783642026768
These are the proceedings of the 18th international conference on domain decomposition methods in science and engineering, heldin Jerusalem, January 12-17, 2008. Domain decomposition methods are iterative methods for solving the often very large linear or nonlinear systems of algebraic equations that arise when various problems in continuum mechanics are discretized using finite elements. They are designed for massively parallel computers and take the memory hierarchy of such systems into account. This is essential for approaching peak floating point performance. There is an increasingly well developed theory which is having a direct impact on the development and improvements of these algorithms.
We are concerned with structural optimization problems where the state variables are supposed to satisfy a PDE or a system of PDEs and the design variables are subject to inequality constraints. Within a primal-dual s...
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ISBN:
(纸本)9783642026775;9783642026768
We are concerned with structural optimization problems where the state variables are supposed to satisfy a PDE or a system of PDEs and the design variables are subject to inequality constraints. Within a primal-dual setting, we suggest an all-at-once approach based on interior-point methods. Coupling the inequality constraints by logarithmic barrier functions involving a barrier parameter and the PDE by Lagrange multipliers, the KKT conditions for the resulting saddle point problem represent a parameter dependent nonlinear system. The efficient numerical solution relies on multilevel path-following predictor-corrector techniques with an adaptive choice of the continuation parameter where the discretization is taken care of by finite elements with respect to nested hierarchies of simplicial triangulations of the computational domain. In particular, the predictor is a nested iteration type tangent continuation, whereas the corrector is a multilevel inexact Newton method featuring transforming null space iterations. As an application in life sciences, we consider the optimal shape design of capillary barriers in microfluidic biochips.
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