Much of uncertainty quantification to date has focused on determining the effect of variables modeled probabilistically, and with a known distribution, on some physical or engineering system. We develop methods to obtain information on the system when the distributions of some variables are known exactly, others are known only approximately, and perhaps others are not modeled as random variables at all.The main tool used is the duality between risk-sensitive integrals and relative entropy, and we obtain explicit bounds on standard performance measures (variances, exceedance probabilities) over families of distributions whose distance from a nominal distribution is measured by relative entropy. The evaluation of the risk-sensitive expectations is based on polynomial chaos expansions, which help keep the computational aspects tractable.

A conceptual numerical strategy for rate-independent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The novelty is that we obtain convergence ofsubsequences of space-time discretizations even in case where the limitproblem does not have a unique solution and we need noadditional assumptions on higher regularity of the limit solution.The variety of general perspectives thus obtained is illustrated on several specific examples: plasticity with isotropic hardening, damage, debonding, magnetostriction, and two models of martensitic transformation in shape-memory alloys.

This paper describes the extension of arecently developed numerical solver for the Landau-LifshitzNavier-Stokes (LLNS) equations to binary mixtures in threedimensions. The LLNS equations incorporate thermal fluctuations intomacroscopic hydrodynamics by using white-noise fluxes. Thesestochastic PDEs are more complicated in three dimensions due to thetensorial form of the correlations for the stochastic fluxes and inmixtures due to couplings of energy and concentration fluxes (e.g.,Soret effect). We present various numerical tests of systems in andout of equilibrium, including time-dependent systems, anddemonstrate good agreement with theoretical results and molecularsimulation.

In this paper, we propose a numerical method to solve stochastic elliptic interfaceproblems with random interfaces. Shape calculus is first employed to derive theshape-Taylor expansion in the framework of the asymptotic perturbation approach. Given themean field and the two-point correlation function of the random interface, we can thusquantify the mean field and the variance of the random solution in terms of certain ordersof the perturbation amplitude by solving a deterministic elliptic interface problem andits tensorized counterpart with respect to the reference interface. Error estimates arederived for the interface-resolved finite element approximation in both, the physical andthe stochastic dimension. In particular, a fast finite difference scheme is proposed tocompute the variance of random solutions by using a low-rank approximation based on thepivoted Cholesky decomposition. Numerical experiments are presented to validate andquantify the method.

We construct a Roe-type numerical scheme for approximating the solutionsof a drift-flux two-phase flow model. The model incorporates a set ofhighly complex closure laws, and the fluxes are generally not algebraic functions of the conserved variables. Hence, the classical approach of constructing a Roe solver by means of parameter vectors is unfeasible.Alternative approaches for analytically constructing the Roe solver are discussed, and a formulation of the Roe solver valid for general closure laws is derived. In particular, a fully analytical Roe matrix is obtained for the special case of the Zuber–Findlay law describing bubbly flows.First and second-order accurate versions of the scheme are demonstrated bynumerical examples.

In this paper, by use of affine biquadratic elements, we constructand analyze a finite volume element scheme for elliptic equations onquadrilateral meshes. The scheme is shown to be of second-order inH 1-norm, provided that each quadrilateral in partition is almosta parallelogram. Numerical experiments are presented to confirm theusefulness and efficiency of the method.

In this paper, we present and study a mixed variational method in order to approximate,with the finite element method, a Stokes problem with Tresca friction boundary conditions.These non-linear boundary conditions arise in the modeling of mold filling process bypolymer melt, which can slip on a solid wall. The mixed formulation is based on adualization of the non-differentiable term which define the slip conditions. Existence anduniqueness of both continuous and discrete solutions of these problems is guaranteed bymeans of continuous and discrete inf-sup conditions that are proved. Velocity and pressureare approximated by P1 bubble-P1 finite element and piecewise linearelements are used to discretize the Lagrange multiplier associated to the shear stress onthe friction boundary. Optimal a priori error estimates are derived usingclassical tools of finite element analysis and two uncoupled discrete inf-sup conditionsfor the pressure and the Lagrange multiplier associated to the fluid shear stress.

We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusionequation on an interval. The discretization is based on the equation’s gradient flowstructure with respect to the Wasserstein distance. The scheme inherits various propertiesfrom the continuous flow, like entropy monotonicity, mass preservation, metric contractionand minimum/ maximum principles. As the main result, we give a proof of convergence in thelimit of vanishing mesh size under a CFL-type condition. We also present results fromnumerical experiments.

The paper deals with the application of a non-conforming domaindecomposition methodto the problem of the computation of induced currents in electric engineswith moving conductors.The eddy currents model is considered as a quasi-staticapproximation of Maxwellequations and we study its two-dimensional formulation with either themodified magnetic vector potential or the magnetic field as primary variable.Two discretizations are proposed, the first one based on curved finiteelementsand the second one based on iso-parametric finite elements in both thestatic and movingparts. The coupling is obtained by means of the mortar element method(see [CITE])and the approximation on the whole domain turns out to be non-conforming.In bothcases optimal error estimates are provided. Numerical tests are then proposed in the case of standard first order finiteelements to test the reliability and precision of the method. An applicationof the method to study the influence of the free part movement on thecurrents distribution is also provided.

In order to accommodate solutions with multiple phases, corresponding to crossing rays, weformulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite and known a priori we can recover the exact multiphase solution from anassociated system of moment equations, closed by an assumption on the form of the density function in the kinetic equation. We consider two different closure assumptions based ondelta and Heaviside functions and analyze the resultingequations. They form systems of nonlinear conservation laws with source terms. In contrast to the classicaleikonal equation, theseequations will incorporate a "finite" superposition principle in the sense that while the maximum number of phasesis not exceeded a sum of solutions is also a solution. We present numerical results for a varietyof homogeneous and inhomogeneous problems.

We consider the lowest-order Raviart–Thomas mixed finite elementmethod for second-order elliptic problems on simplicial meshes intwo and three space dimensions. This method produces saddle-pointproblems for scalar and flux unknowns. We show how to easily andlocally eliminate the flux unknowns, which implies the equivalencebetween this method and a particular multi-point finite volumescheme, without any approximate numerical integration. The matrixof the final linear system is sparse, positive definite for alarge class of problems, but in general nonsymmetric. We next showthat these ideas also apply to mixed and upwind-mixed finiteelement discretizations of nonlinear parabolicconvection–diffusion–reaction problems. Besides the theoreticalrelationship between the two methods, the results allow forimportant computational savings in the mixed finite elementmethod, which we finally illustrate on a set of numericalexperiments.

The topic of this work is to obtain discrete Sobolev inequalities for piecewise constant functions, and to deduce Lp error estimates on the approximate solutions of convection diffusion equations by finite volume schemes.

A distributed-parameter (one-dimensional) anatomically detailed model for the arterialnetwork of the arm is developed in order to carry out hemodynamics simulations. This workfocuses on the specific aspects related to the model set-up. In this regard, stringentanatomical and physiological considerations have been pursued in order to construct thearterial topology and to provide a systematic estimation of the involved parameters. Themodel comprises 108 arterial segments, with 64 main arteries and 44 perforator arteries,with lumen radii ranging from 0.24 cm – axillary artery- to 0.018 cm – perforatorarteries. The modeling of blood flow in deformable vessels is governed by a well-known setof hyperbolic partial differential equations that accounts for mass and momentumconservation and a constitutive equation for the arterial wall. The variationalformulation used to solve the problem and the related numerical approach are described.The model rendered consistent pressure and flow rate outputs when compared with patientrecords already published in the literature. In addition, an application todimensionally-heterogeneous modeling is presented in which the developed arterial networkis employed as an underlying model for a three-dimensional geometry of a branching pointto be embedded in order to perform local analyses.

We consider a model for flow in a porous medium with a fracture in which the flow in thefracture is governed by the Darcy−Forchheimerlaw while that in the surrounding matrix is governed byDarcy’s law. We give an appropriate mixed, variational formulation and show existence anduniqueness of the solution. To show existence we give an analogous formulation for themodel in which the Darcy−Forchheimerlaw is the governing equation throughout the domain. We showexistence and uniqueness of the solution and show that the solution for the model withDarcy’s law in the matrix is the weak limit of solutions of the model with theDarcy−Forchheimerlaw in theentire domain when the Forchheimer coefficient in the matrix tends toward zero.

We illustrate how some interesting new variational principles can beused for the numerical approximation of solutions to certain (possiblydegenerate) parabolic partial differential equations. One remarkablefeature of the algorithms presented here is that derivatives do notenter into the variational principles, so, for example, discontinuousapproximations may be used for approximating the heat equation. Wepresent formulae for computing a Wasserstein metric which entersinto the variational formulations.

Since matrix compression has paved the way for discretizing the boundary integralequation formulations of electromagnetics scattering on very fine meshes, preconditionersfor the resulting linear systems have become key to efficient simulations. Operatorpreconditioning based on Calderón identities has proved to be a powerful device fordevising preconditioners. However, this is not possible for the usual first-kind boundaryformulations for electromagnetic scattering at general penetrable composite obstacles. Wepropose a new first-kind boundary integral equation formulation following the reasoningemployed in [X. Clayes and R. Hiptmair, Report 2011-45, SAM, ETH Zürich (2011)] foracoustic scattering. We call it multi-trace formulation, because itsunknowns are two pairs of traces on interfaces in the interior of the scatterer. We give acomprehensive analysis culminating in a proof of coercivity, and uniqueness and existenceof solution. We establish a Calderón identity for the multi-trace formulation, which formsthe foundation for operator preconditioning in the case of conforming Galerkin boundaryelement discretization.

We consider the identification of a distributed parameter in an ellipticvariational inequality. On the basis of an optimal control problemformulation, the application of a primal-dual penalizationtechnique enables us to prove the existenceof multipliers giving a first order characterization of the optimal solution.Concerning the parameter we consider differentregularity requirements. For the numerical realization we utilize a complementarity function,which allows us to rewrite the optimality conditions as a set of equalities.Finally, numerical results obtained from a least squares type algorithmemphasize the feasibility of our approach.

A mixed finite element method for the Navier–Stokes equations is introduced in which thestress is a primary variable. The variational formulation retains the mathematicalstructure of the Navier–Stokes equations and the classical theory extends naturally tothis setting. Finite element spaces satisfying the associated inf–sup conditions aredeveloped.

This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C 1,1 boundary. It is an extension of anearlier result of [Phung, ESAIM: COCV9 (2003) 621–635] for domains of class C ∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces a nearly optimal convergence rate for the method of quasi-reversibility introduced in [Lattès and Lions, Dunod (1967)] to solve the ill-posed Cauchy problems.

Motivated by the pricing of American options for baskets weconsider a parabolic variational inequality in a boundedpolyhedral domain $\Omega\subset\mathbb{R}^d$ with a continuous piecewisesmooth obstacle. We formulate a fully discrete method by usingpiecewise linear finite elements in space and the backward Eulermethod in time. We define an a posteriori error estimator and showthat it gives an upper bound for the error inL2(0,T;H1 (Ω)). The error estimator is localized in thesense that the size of the elliptic residual is only relevant inthe approximate non-contact region, and the approximability of theobstacle is only relevant in the approximate contact region. Wealso obtain lower bound results for the space error indicators inthe non-contact region, and for the time error estimator.Numerical results for d=1,2 show that the error estimator decayswith the same rate as the actual error when the space meshsize hand the time step τ tend to zero. Also, the error indicatorscapture the correct behavior of the errors in both the contact andthe non-contact regions.