This paper presents a multilevel algorithm to accelerate the numerical solution of thin shell finite element problems described by subdivision surfaces. Subdivision surfaces have become a widely used geometric representation for general curved three dimensional boundary models and thin shells as they provide a compact and robust framework for modeling 3D geometry. More recently, the shape functions used in the subdivision surfaces framework have been proposed as candidates for use as finite element basis functions in the analysis and simulation of the mechanical deformation of thin shell structures. When coupled with standard solvers, however, such simulations do not scale well. Run time costs associated with high-resolution simulations (10^5 degrees of freedom or more) become prohibitive. The main

Incompressible interfacial flows here refer to those incompressible flows possessing multiple distinct, immiscible fluids separated by interfaces of arbitrarily complex topology. A prototypical example is free surface flows, where fluid properties across the interface vary by orders of magnitude. Interfaces present in these flows possess topologies that are not only irregular but also dynamic, undergoing gross changes such as merging, tearing, and filamenting as a result of the flow and interface physics such as surface tension and phase change. The interface topology requirements facing an algorithm tasked to model these flows inevitably leads to an underlying Eulerian methodology. The discussion herein is confined therefore to Eulerian schemes, with further emphasis on finite volume methods of discretization for the partial differential equations manifesting the physical model. Numerous algorithm choices confront users and developers of simulation tools designed to model the time-un...

Abstract. In this article we develop and analyze two-level and multi-level methods for the family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with rough coefficients (exhibiting large jumps across interfaces in the domain). These methods are based on a decomposition of the DG finite element space that inherently hinges on the diffusion coefficient of the problem. Our analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes, and we establish both robustness with respect to the jump in the coefficient and near-optimality with respect to the mesh size. Following the analysis, we present a sequence of detailed numerical results which verify the

The remarkable robustness of Krylov methods with respect to inexact matrixvector products is a strong property recently emphasised [11, 2, 3]. In the context of embedded iterative solvers with an outer Krylov scheme, it is possible to monitor the inner accuracy and relax it when outer convergence proceeds. We extend the relaxation strategy proposed in [2] to the context of domain decomposition methods for partial dierential equations solved with the Schur complement method. Numerical experiments on an heterogeneous and anisotropic problem show that it is possible to save a signicant amount of matrix-vector products when using a tuned relaxation strategy for controlling the accuracy of the local subproblems. 1 Introduction Iterative processes are widely used in Linear Algebra for treating large sets of data. It becomes more and more common now that one iterative solver has to be embedded in an outer one: this is the case, for instance, for solving eigenproblems with inverse iteration...

Summary The goal of this paper is to design optimal multilevel solvers for the finite element approximation of second order linear elliptic problems with piecewise constant coefficients on bisection grids. Local multigrid and BPX preconditioners are constructed based on local smoothing only at the newest vertices and their immediate neighbors. The analysis of eigenvalue distributions for these local multilevel preconditioned systems shows that there are only a fixed number of eigenvalues which are deteriorated by the large jump. The remaining eigenvalues are bounded uniformly with respect to the coefficients and the meshsize. Therefore, the resulting preconditioned conjugate gradient algorithm will converge with an asymptotic rate independent of the coefficients and logarithmically with respect to the meshsize. As a result, the overall computational complexity is nearly optimal.

Abstract. In this article we develop and analyze two-level and multi-level methods for the family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with rough coefficients (exhibiting large jumps across interfaces in the domain). These methods are based on a decomposition of the DG finite element space that inherently hinges on the diffusion coefficient of the problem. Our analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes, and we establish both robustness with respect to the jump in the coefficient and near-optimality with respect to the mesh size. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods. Contents