We present analytical solution of the Stokes problem in rotationally symmetric domains. This is then used to find the asymptotic behaviour of the solution in the vicinity of corners, also for Navier-Stokes equations. We apply this to construct very precise numerical finite element solution.
Balancing Domain Decomposition by Constraints (BDDC) belongs to the class of primal substructuring Domain Decomposition (DD) methods. DD methods are iterative methods successfully used in engineering to parallelize solution of large linear systems arising from discretization of second order elliptic problems. Substructuring DD methods represent an important class of DD methods. Their main idea is to divide the underlying domain into nonoverlapping subdomains and solve many relatively small, local...
We describe a parallel implementation of the Balancing Domain Decomposition by Constraints (BDDC) method enhanced by an adaptive construction of coarse problem. The method is designed for numerically difficult problems, where standard choice of continuity of arithmetic averages across faces and edges of subdomains fails to maintain the low condition number of the preconditioned system. Problems of elasticity analysis of bodies consisting of different materials with rapidly changing stiffness may...
We present analytical solution of the Stokes problem in 2D domains. This is then used to find the asymptotic behavior of the solution in the vicinity of corners, also for Navier-Stokes equations in 2D. We apply this to construct very precise numerical finite element solution.
In computer fluid dynamics, employing stabilization to the finite element method is a commonly accepted way to improve the applicability of this method to high Reynolds numbers. Although the accompanying loss of accuracy is often referred, the question of quantifying this defect is still open. On the other hand, practitioners call for measuring the error and accuracy. In the paper, we present a novel approach for quantifying the difference caused by stabilization.
Different choices of the averaging operator within the BDDC method are compared on a series of 2D experiments. Subdomains with irregular interface and with jumps in material coefficients are included into the study. Two new approaches are studied along three standard choices. No approach is shown to be universally superior to others, and the resulting recommendation is that an actual method should be chosen based on properties of the problem.
In this paper, we discuss the choice of weights in averaging of local (subdomain) solutions on the interface for the BDDC method (Balancing Domain Decomposition by Constraints). We try to find relations among different choices of the interface weights and compare them numerically on model problems of the Poisson equation and linear elasticity in 3D. Problems with jumps in coefficients of material properties are considered and both regular and irregular interfaces between subdomains are tested.
We deal with numerical simulation of incompressible flow governed by the Navier-Stokes equations. The problem is discretised using the finite element method, and the arising system of nonlinear equations is solved by Picard iteration. We explore the applicability of the Balancing Domain Decomposition by Constraints (BDDC) method to nonsymmetric problems arising from such linearisation. One step of BDDC is applied as the preconditioner for the stabilized variant of the biconjugate gradient (BiCGstab)...
We consider the Navier-Stokes equations for the incompressible flow in channels with forward and backward steps. The paper consists of two main parts. In the first part we investigate a posteriori error estimates for the Stokes and Navier-Stokes equations on two-dimensional polygonal domains. We apply the a posteriori estimates to solve an incompressible flow problem in a domain with corners that cause singularities in the solution. Second part of the paper stands on the result on the asymptotics of...
We deal with modelling of flows in channels or tubes with abrupt changes of the diameter. The goal of this work is to construct the FEM solution in the vicinity of these corners as precise as desired. We present two ways. The first approach makes use of a posteriori error estimates and the adaptive strategy. The second approach is based on the asymptotic behaviour of the exact solution in the vicinity of the corner and on the a priori error estimate of the FEM solution. Then we obtain the solution...
In this paper, we introduce a general framework for derivation of the averaging operator, from which the standard choices are recovered by simplifications. Then, an alternative approach derived by another simplification is proposed and tested on a 2D example.
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