383 research outputs found
A p-multigrid method enhanced with an ILUT smoother and its comparison to h-multigrid methods within Isogeometric Analysis
Over the years, Isogeometric Analysis has shown to be a successful
alternative to the Finite Element Method (FEM). However, solving the resulting
linear systems of equations efficiently remains a challenging task. In this
paper, we consider a p-multigrid method, in which coarsening is applied in the
approximation order p instead of the mesh width h. Since the use of classical
smoothers (e.g. Gauss-Seidel) results in a p-multigrid method with
deteriorating performance for higher values of p, the use of an ILUT smoother
is investigated. Numerical results and a spectral analysis indicate that the
resulting p-multigrid method exhibits convergence rates independent of h and p.
In particular, we compare both coarsening strategies (e.g. coarsening in h or
p) adopting both smoothers for a variety of two and threedimensional
benchmarks
A comparison of Krylov methods for Shifted Skew-Symmetric Systems
It is well known that for general linear systems, only optimal Krylov methods
with long recurrences exist. For special classes of linear systems it is
possible to find optimal Krylov methods with short recurrences. In this paper
we consider the important class of linear systems with a shifted skew-symmetric
coefficient matrix. We present the MRS3 solver, a minimal residual method that
solves these problems using short vector recurrences. We give an overview of
existing Krylov solvers that can be used to solve these problems, and compare
them with the MRS3 method, both theoretically and by numerical experiments.
From this comparison we argue that the MRS3 solver is the fastest and most
robust of these Krylov method for systems with a shifted skew-symmetric
coefficient matrix.Comment: 23 pages, 3 figure
A mathematical model for the dissolution of particles in multi-component alloys
Dissolution of stoichiometric multi-component particles is an important process ocurring during the heat treatment of as-cast aluminium alloys prior to hot extrusion. A mathematical model is proposed to describe such a process. In this model equations are given to determine the position of the particle interface in time, using a number of diffusion equations which are coupled by nonlinear boundary conditions at the interface. This problem is known as a vector valued Stefan problem. Moreover the well-posedness of the moving boundary problem is investigated using the maximum principle for the parabolic partial differential equation. Furthermore, for an unbounded domain and planar co-ordinates an analytical asymptotic approximation based on self-similarity is derived. Moreover, this self-similar solution and the asymptotic approximation are extended to the vector valued Stefan problem. The approaches are compared to each other and the asymptotic approximation is used to gain insight into the influence of all components on the dissolution. Subsequently a numerical treatment of the vector valued Stefan problem is described. The numerical method is compared with solutions by analytical methods. Finally, an example is shown
A mathematical model for the dissolution of particles in multi-component alloys
AbstractDissolution of stoichiometric multi-component particles is an important process occurring during the heat treatment of as-cast aluminum alloys prior to hot extrusion. A mathematical model is proposed to describe such a process. In this model equations are given to determine the position of the particle interface in time, using a number of diffusion equations which are coupled by nonlinear boundary conditions at the interface. This problem is known as a vector valued Stefan problem. A necessary condition for existence of a solution of the moving boundary problem is proposed and investigated using the maximum principle for the parabolic partial differential equation. Furthermore, for an unbounded domain and planar co-ordinates an asymptotic approximation based on self-similarity is derived. The asymptotic approximation is used to gain insight into the influence of all components on the dissolution. Subsequently, a numerical treatment of the vector valued Stefan problem is described. The numerical solution is compared with solutions obtained by the analytical methods. Finally, an example is shown
A vector valued Stefan problem from aluminium industry
Dissolution of stoichiometric multi-component particles in ternary alloys is an important process occurring during the heat treatment of as-cast aluminium alloys prior to hot-extrusion. A mathematical model is proposed to describe such a process. In this model an equation is given to determine the position of the particle interface in time, using two diffusion equations which are coupled by nonlinear boundary conditions at the interface. Moreover the well-posedness of the moving boundary problem is investigated using the maximum principle for the parabolic partial differential equation. Furthermore, for an unbounded domain and planar co-ordinates an analytical asymptotic approximation based on self-similarity is derived. This asymptotic approximation gives insight into the well-posedness of the problem
www.elsevier.com/locate/cam On deflation and singular symmetric positive semi-definite matrices
For various applications, it is well-known that the deflated ICCG is an efficient method for solving linear systems with invertible coefficient matrix. We propose two equivalent variants of this deflated ICCG which can also solve linear systems with singular coefficient matrix, arising from discretization of the discontinuous Poisson equation with Neumann boundary conditions. It is demonstrated both theoretically and numerically that the resulting methods accelerate the convergence of the iterative process. Moreover, in practice the singular coefficient matrix has often been made invertible by modifying the last element, since this can be advantageous for the solver. However, the drawback is that the condition number becomes worse-conditioned. We show that this problem can completely be remedied by applying the deflation technique with just one deflation vector
- …