MOX Reports
The preprint collection of the Laboratory for Modeling and Scientific Computation MOX. It mainly contains works on numerical
analysis and mathematical modeling applied to engineering problems. MOX web site is mox.polimi.it
Found 1238 products
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39/2010 - 11/16/2010
D'Angelo, C.
Finite Element Approximation of Elliptic Problems with Dirac Measure Terms in Weighted Spaces. Applications to 1D-3D Coupled Problems | Abstract | | In this work we study the stability and the convergence rates of the finite element approximation of elliptic problems involving Dirac measures, using weighted Sobolev spaces and weighted discrete norms. Our approach handles
both the cases where the measure is simply a right hand side or it represents an additional term, i.e. solution-dependent, in the formulation of the problem.
The main motivation of this study is to provide a methodological tool to treat elliptic problems in fractured domains, where the coupling terms are seen as Dirac measures concentrated on the fractures. We first establish a decomposition lemma, which is our fundamental tool for the analysis of the considered problems in the non-standard setting of weighted spaces. Then, we consider the stability of the Galerkin approxima-
tion with finite elements in weighted norms, with uniform and graded meshes.
We introduce a discrete decomposition lemma that extends the continuous one and allows to derive discrete inf-sup conditions in weighted norms. Then, we focus on the convergence of the finite element method. Due to the lack of regularity, the convergence rates are suboptimal for uniform meshes; we show that using graded meshes optimal rates are recovered. Our theoretical results are supported by several numerical experiments. Finally, we show how our theoretical results apply to certain coupled problems involving
fluid flow in porous three-dimensional media with one-dimensional fractures, that are found in the analysis of microvascular flows.
Keywords: elliptic problems, measure, Dirac measure, weighted spaces, nite element method, graded
mesh, error estimates, reduced models, multiscale models, microcirculation. |
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38/2010 - 11/15/2010
Flournoy, N.; May, C.; Secchi, P.
Response-adaptive designs in clinical trials for targeting the best treatment: an overview | Abstract | | Response-adaptive designs are increasingly being implemented in clinical trials, particularly early phase trials, and they have increasingly stimulated the work of researchers. This paper reviews a particular class of response-adaptive designs, which have a di fferent property from the most adaptive designs in literature. These
are response-adaptive designs targeting asymptotically the superior response, that is, treating with the superior treatment with
probability converging to one. The model underlying such designs is a randomly reinforced urn. In the context of clinical trials, this prop-
erty is particularly attractive from an ethical point of view. This overview starts from the early paper of [8] until the recent work by [9].
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37/2010 - 11/12/2010
Discacciati, Marco; Quarteroni, Alfio; Quinodoz, Samuel
Numerical approximation of internal discontinuity interface problems | Abstract | | This work focuses on the finite element discretization of boundary value problems whose solution presents either a discontinuity and/or a discontinuous conormal derivative across an interface inside the computational domain.
The interface is characterized via a level-set function. The discontinuities are accounted for using suitable extension operators whose numerical
implementation requires a very low computational
e ffort. Numerical results to validate our approach are presented in one, two and three dimensions. |
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36/2010 - 11/11/2010
Arioli, G.; Koch, H.
Non-Symmetric low-index solutions for a symmetric boundary value problem | Abstract | | We consider the equation -Laplacian(u)=w*u^3 on a square domain in R^2, with Dirichlet boundary conditions, where w is a given positive function that is invariant under all (Euclidean) symmetries of the square. This equation is shown to have a solution u, with Morse index 2, that is neither symmetric nor antisymmetric with respect to any nontrivial symmetry of the square. Part of our proof is computer-assisted. An analogous result is proved for index 1.
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35/2010 - 11/10/2010
Arioli, G.; Koch, H.
Integration of dissipative PDEs: a case study | Abstract | | We develop a computer-assisted technique to construct and analyze orbits of dissipative evolution equations. As a case study, the methods are applied to the Kuramoto-Sivashinski equation. We prove the existence of a hyperbolic periodic orbit.
Keywords: Kuramoto-Sivashinski equation, hyperbolicity, periodic orbit, computer-assisted proof
AMS Subject Classification: 37L05, 37L45, 35K35 |
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34/2010 - 11/09/2010
Abba', A.; Bonaventura, L.
A mimetic finite difference method for Large Eddy Simulation of incompressible flow | Abstract | | A finite difference discretization of the three-dimensional, incompressible Navier-Stokes equations is presented, based on finite difference operators that satisfy discrete analogs of some basic calculus identities. These
mimetic properties yield a numerical method for which a discrete form of the vorticity equation can be derived naturally from the discrete momentum equation, by application of the mimetic rotation operator. As a result,a discrete approximation of vorticity is exactly preserved, for inviscid flows, independently of the mesh size. The vorticity preservation property guarantees that no spurious vorticity is generated by the nonlinear advective terms in absence of viscosity. A mimetic discretization of the viscous terms and an appropriate treatment for rigid wall boundary conditions are also
proposed. The relationship of this approach to other similar techniques is discussed. The proposed method is validated on several idealized test cases for laminar incompressible flow, in which it is compared to a widely used finite difference discretization. The method is then applied to Large Eddy Simulation of incompressible flow, demonstrating the advantages of the inherentconservation properties in a comparison with experimental data and DNS results especially when strong vorticity production takes place at the boundaries. |
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33/2010 - 11/08/2010
Migliorati, Giovanni; Quarteroni, Alfio
Multilevel Schwarz Methods for Elliptic Partial Differential Equations | Abstract | | We investigate multilevel Schwarz domain decomposition preconditioners, to efficiently solve linear systems arising from numerical discretizations of elliptic Partial Di fferential Equations by the fiite element method. In our
analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner. We start from two-level preconditioners with either aggregative or interpolative coarse level
components, then we focus on a strategy to increase the number of levels. For all preconditioners, we consider the additive residual update and its multiplicative variants within and between levels. Moreover, we compare the preconditioners behaviour, regarding scalability and rate of convergence. Numerical results are provided for elliptic boundary-value problems, including a convection-di ffusion problem when suitable stabilization
becomes necessary. |
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32/2010 - 11/07/2010
Malossi, A. Cristiano I.; Blanco, Pablo J.; Deparis, Simome; Quarteroni, Alfio
Algorithms for the partitioned solution of weakly coupled fluid models for cardiovascular flows | Abstract | | The main goal of the present work is to devise robust iterative strategies to partition the solution of the Navier–Stokes equations in a three-dimensional(3D) computational domain, into non overlapping 3D subdomains,which communicate through the exchange of integrated quantities
across the interfaces. The novel aspect of the present approach is that at coupling boundaries the conservation of flow rates and of the associated dual variables is imposed, entailing a weak physical coupling. For the solution
of the non-linear problem, written in terms of interfaces variables, two strategies are compared: relaxed fixed point iterations and Newton iterations. The algorithm is tested in several configurations for problems which involve more than two components at each coupling interface. In such cases it is shown that relaxed fixed point methods are not convergent, whereas the Newton method leads in all the tested cases to convergent schemes. One of the appealing aspects of the strategy proposed here is the
flexibility in the setting of boundary conditions at branching points, where no hierarchy is established a priori, unlike classical Gauss–Seidel methods.
Such an approach can be applied in two other different contexts: (i) when coupling dimensionally-heterogeneous models, just by replacing some of the 3D models by one-dimensional (or zero-dimensional) condensed ones, and (ii) as a preconditioner method for domain decomposition methods for the
Navier–Stokes equations. These two issues are also addressed in the present work. Finally, several examples of application are presented, ranging from academic examples to some related to the computational hemodynamics field. |
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