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
-
12/2015 - 03/16/2015
Antonietti, P. F.; Beirao da Veiga, L.; Scacchi, S.; Verani, M.
A $C^1$ virtual element method for the Cahn-Hilliard equation with polygonal meshes | Abstract | | In this paper we develop an evolution of the $C^1$ virtual elements of minimal degree for the approximation of the Cahn-Hilliard equation.
The proposed method has the advantage of being conforming in $H^2$ and making use of a very simple set of degrees of freedom, namely 3 degrees of freedom per vertex of the mesh. Moreover, although the present method is new also on triangles, it can make use of general polygonal meshes. As a theoretical and practical support, we prove the convergence of the semi-discrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests. |
-
11/2015 - 03/05/2015
Antonietti, P. F.; Marcati, C.; Mazzieri, I.; Quarteroni, A.
High order discontinuous Galerkin methods on simplicial elements for the elastodynamics equation | Abstract | | In this work apply the discontinuous Galekin (dG) spectral element method on meshes made of simplicial elements for the approximation of the elastodynamics equation. Our approach combines the high accuracy of spectral methods, the geometrical flexibility of simplicial elements and the computational flexibility of dG methods. We analyze the dissipation, dispersion and stability properties of the resulting scheme, with a focus on the choice of different sets of basis functions. Finally, we test the
method on benchmark as well as realistic test cases. |
-
10/2015 - 02/12/2015
Antonietti, P. F.; Grasselli, M.; Stangalino, S.; Verani, M.
Discontinuous Galerkin approximation of linear parabolic problems with dynamic boundary conditions | Abstract | | In this paper we propose and analyze a Discontinuous Galerkin method for a
linear parabolic problem with dynamic boundary conditions. We present the formulation and prove stability and optimal a priori error estimates for the fully discrete scheme. More precisely, using polynomials of degree $pgeq 1$ on meshes with granularity $h$ along with a backward Euler time-stepping scheme with time-step $Delta t$, we prove that the fully-discrete solution is bounded by the data and it converges, in a suitable (mesh-dependent) energy norm, to the exact solution with optimal order $h^p + Delta t$. The sharpness of the theoretical estimates are verified through several numerical experiments. |
-
09/2015 - 02/11/2015
Ghiglietti, A.; Ieva, F.; Paganoni, A.M.; Aletti, G.
On linear regression models in infinite dimensional spaces with scalar response | Abstract | | In functional linear regression, the parameters estimation involves solving a non necessarily well-posed problem and it has points of contact with a range of methodologies, including statistical smoothing, deconvolution and projection on finite-dimensional subspaces. We discuss the standard approach based explicitly on functional principal components analysis, nevertheless the choice of the number of basis components remains something subjective and not always properly discussed and justified. In this work we discuss inferential properties of least square estimation in this context with different choices of projection subspaces, as well as we study asymptotic behaviour increasing the dimension of subspaces. |
-
06/2015 - 02/11/2015
Perotto, S.; Zilio, A.
Space-time adaptive hierarchical model reduction for parabolic equations | Abstract | | We formalize the pointwise HiMod approach in an unsteady setting,
by resorting to a model discontinuous in time, continuous and hierarchically reduced in space.
The selection of the modal distribution and of the space-time discretization is automatically performed
via an a posteriori analysis of the global error.
The results of the numerical verification confirm the robustness of the proposed adaptive procedure
in terms of accuracy as well as of sensitivity with respect to the goal quantity.
The validation results in the groundwater experimental setting are actually more than satisfying,
with an improvement in the concentration predictions by means of the adaptive HiMod approximation. |
-
07/2015 - 02/05/2015
Giovanardi, B.; Scotti, A.; Formaggia, L.; Ruffo, P.
A general framework for the simulation of geochemical compaction | Abstract | | We propose a mathematical model and a numerical scheme to describe compaction
processes in a sedimentary rock layer undergoing both mechanical and
geochemical processes. We simulate the sedimentation process by providing a
sedimentation rate and we account for chemical reactions using simplified kinetics
describing either the conversion of a solid matrix into a fluid, as in the case of
kerogen degradation into oil, or the precipitation of a mineral solute on the solid
matrix of the rock. We use a Lagrangian description that enables to recast the
equations in a fixed frame of reference. We present an iterative splitting scheme
that allows solving the set of governing equations efficiently in a sequential manner.
We assess the performances of this strategy in terms of convergence and mass
conservation. Some numerical experiments show the capability of the scheme to
treat two test cases, one concerning the precipitation of a mineral, the other the
dissolution of kerogen. |
-
08/2015 - 02/02/2015
Agosti, A.; Formaggia, L.; Giovanardi, B.; Scotti, A.
Numerical simulation of geochemical compaction with discontinuous reactions | Abstract | | The present work deals with the numerical simulation of porous media subject to the coupled effects of mechanical compaction and reactive flows that can significantly alter the porosity due to dissolution, precipitation or transformation of the solid matrix. These chemical processes can be effectively modelled by ODEs with discontinuous right hand side, where the discontinuity depends on time and on the solution itself. Filippov theory can be applied to prove existence and to determine the solution behaviour at the discontinuities. From the numerical point of view, tailored numerical schemes are needed to guarantee positivity, mass conservation and accuracy. In particular, we rely on an event-driven approach such that, if the trajectory crosses a discontinuity, the transition point is localized exactly and integration is restarted accordingly.
|
-
05/2015 - 01/29/2015
Chen, P.; Quarteroni, A.; Rozza, G.
Reduced order methods for uncertainty quantification problems | Abstract | | This work provides a review on reduced order methods in solving uncertainty quantification problems. A quick introduction of the reduced order methods, including proper orthogonal decomposition and greedy reduced basis methods, are presented along with the essential components of general greedy algorithm, a posteriori error estimation and Offline-Online decomposition. More advanced reduced order methods are then developed for solving typical uncertainty quantification problems involving pointwise evaluation and/or statistical integration, such as failure probability evaluation, Bayesian inverse problems and variational data assimilation. Three expository examples are provided to demonstrate the efficiency and accuracy of the reduced order methods, shedding the light on their potential for solving problems dealing with more general outputs, as well as time dependent, vectorial noncoercive parametrized |
|
