Subsurface flows are influenced by the presence of faults and large fractures which actas preferential paths or barriers for the flow. In literature models were proposed tohandle fractures in a porous medium as objects of codimension 1. In this work we considerthe case of a network of intersecting fractures, with the aim of deriving physicallyconsistent and effective interface conditions to impose at the intersection betweenfractures. This new model accounts for the angle between fractures at the intersectionsand allows for jumps of pressure across intersections. This fact permits to describe theflow when fractures are characterized by different properties more accurately with respectto other models that impose pressure continuity. The main mathematical properties of themodel, derived in the two-dimensional setting, are analyzed. As concerns the numericaldiscretization we allow the grids of the fractures to be independent, thus in generalnon-matching at the intersection, by means of the extended finite element method(XFEM). This increases the flexibility of the method in the case of complexgeometries characterized by a high number of fractures.

We present in this paper a proof of well-posedness and convergence for the parallelSchwarz Waveform Relaxation Algorithm adapted to an N-dimensional semilinear heatequation. Since the equation we study is an evolution one, each subproblem at each stephas its own local existence time, we then determine a common existence time for everyproblem in any subdomain at any step. We also introduce a new technique: Exponential DecayError Estimates, to prove the convergence of the Schwarz Methods, with multisubdomains,and then apply it to our problem.

We present equivalent conditions and asymptotic models for the diffraction problem ofelastic and acoustic waves in a solid medium surrounded by a thin layer of fluid medium.Due to the thinness of the layer with respect to the wavelength, this problem is wellsuited for the notion of equivalent conditions and the effect of the fluid medium on thesolid is as a first approximation local. We derive and validate equivalent conditions upto the fourth order for the elastic displacement. These conditions approximate theacoustic waves which propagate in the fluid region. This approach leads to solve onlyelastic equations. The construction of equivalent conditions is based on a multiscaleexpansion in power series of the thickness of the layer for the solution of thetransmission problem.

The chemical master equation is a fundamental equation in chemical kinetics. It underliesthe classical reaction-rate equations and takes stochastic effects into account. In thispaper we give a simple argument showing that the solutions of a large class of chemicalmaster equations are bounded in weighted ℓ1-spaces and possess high-ordermoments. This class includes all equations in which no reactions between two or morealready present molecules and further external reactants occur that add mass to thesystem. As an illustration for the implications of this kind of regularity, we analyze theeffect of truncating the state space. This leads to an error analysis for the finite stateprojections of the chemical master equation, an approximation that forms the basis of manynumerical methods.

We address multiscale elliptic problems with random coefficients that are a perturbationof multiscale deterministic problems. Our approach consists in taking benefit of theperturbative context to suitably modify the classical Finite Element basis into adeterministic multiscale Finite Element basis. The latter essentially shares the sameapproximation properties as a multiscale Finite Element basis directly generated on therandom problem. The specific reference method that we use is the Multiscale Finite ElementMethod. Using numerical experiments, we demonstrate the efficiency of our approach and thecomputational speed-up with respect to a more standard approach. In the stationarysetting, we provide a complete analysis of the approach, extending that available for thedeterministic periodic setting.

We propose a multiscale model reduction method for partial differential equations. Themain purpose of this method is to derive an effective equation for multiscale problemswithout scale separation. An essential ingredient of our method is to decompose theharmonic coordinates into a smooth part and a highly oscillatory part so that the smoothpart is invertible and the highly oscillatory part is small. Such a decomposition plays akey role in our construction of the effective equation. We show that the solution to theeffective equation is in H2, and can be approximated by a regularcoarse mesh. When the multiscale problem has scale separation and a periodic structure,our method recovers the traditional homogenized equation. Furthermore, we provide erroranalysis for our method and show that the solution to the effective equation is close tothe original multiscale solution in the H1 norm. Numerical results are presentedto demonstrate the accuracy and robustness of the proposed method for several multiscaleproblems without scale separation, including a problem with a high contrastcoefficient.

The reduced basis method is a model reduction technique yielding substantial savings ofcomputational time when a solution to a parametrized equation has to be computed for manyvalues of the parameter. Certification of the approximation is possible by means of ana posteriori error bound. Under appropriate assumptions, this errorbound is computed with an algorithm of complexity independent of the size of the fullproblem. In practice, the evaluation of the error bound can become very sensitive toround-off errors. We propose herein an explanation of this fact. A first remedy has beenproposed in [F. Casenave, Accurate a posteriori error evaluation in thereduced basis method. C. R. Math. Acad. Sci. Paris 350(2012) 539–542.]. Herein, we improve this remedy by proposing a new approximationof the error bound using the empirical interpolation method (EIM). This method achieveshigher levels of accuracy and requires potentially less precomputations than the usualformula. A version of the EIM stabilized with respect to round-off errors is also derived.The method is illustrated on a simple one-dimensional diffusion problem and athree-dimensional acoustic scattering problem solved by a boundary element method.

The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization error is analyzed for the piecewise linear approximation in the H1(Ω)- and L2(Ω)-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L2(Ω)-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equations and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes.

We extend our results on fictitious domain methods for Poisson’s problem to the case of incompressible elasticity, or Stokes’ problem. The mesh is not fitted to the domain boundary. Instead boundary conditions are imposed using a stabilized Nitsche type approach. Control of the non-physical degrees of freedom, i.e., those outside the physical domain, is obtained thanks to a ghost penalty term for both velocities and pressures. Both inf-sup stable and stabilized velocity pressure pairs are considered.

In this paper, we propose a method for the approximation of the solution ofhigh-dimensional weakly coercive problems formulated in tensor spaces using low-rankapproximation formats. The method can be seen as a perturbation of a minimal residualmethod with a measure of the residual corresponding to the error in a specified solutionnorm. The residual norm can be designed such that the resulting low-rank approximationsare optimal with respect to particular norms of interest, thus allowing to take intoaccount a particular objective in the definition of reduced order approximations ofhigh-dimensional problems. We introduce and analyze an iterative algorithm that is able toprovide an approximation of the optimal approximation of the solution in a given low-ranksubset, without any a priori information on this solution. We alsointroduce a weak greedy algorithm which uses this perturbed minimal residual method forthe computation of successive greedy corrections in small tensor subsets. We prove itsconvergence under some conditions on the parameters of the algorithm. The proposednumerical method is applied to the solution of a stochastic partial differential equationwhich is discretized using standard Galerkin methods in tensor product spaces.

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.

This paper focuses on a one-dimensional wave equation being subjected to a unilateralboundary condition. Under appropriate regularity assumptions on the initial data, a newproof of existence and uniqueness results is proposed. The mass redistribution method,which is based on a redistribution of the body mass such that there is no inertia at thecontact node, is introduced and its convergence is proved. Finally, some numericalexperiments are reported.

In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems [Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Submitted.; Y. Efendiev, J. Galvis and X. Wu, J. Comput. Phys. 230 (2011) 937–955; J. Galvis and Y. Efendiev, SIAM Multiscale Model. Simul. 8 (2010) 1461–1483.] is extended to treat a class of nonlinear elliptic operators. The proposed method requires the solutions of (small dimension and local) nonlinear eigenvalue problems in order to systematically enrich the coarse solution space. Convergence of the method is shown to relate to the dimension of the coarse space (due to the enrichment procedure) as well as the coarse mesh size. In addition, it is shown that the coarse mesh spaces can be effectively used in two-level domain decomposition preconditioners. A number of numerical results are presented to complement the analysis.

In this paper we deal with a semilinear hyperbolic chemotaxis model in one space dimension evolving on a network, with suitable transmission conditions at nodes. This framework is motivated by tissue-engineering scaffolds used for improving wound healing. We introduce a numerical scheme, which guarantees global mass densities conservation. Moreover our scheme is able to yield a correct approximation of the effects of the source term at equilibrium. Several numerical tests are presented to show the behavior of solutions and to discuss the stability and the accuracy of our approximation.

In this paper, we propose implicit and semi-implicit in time finite volume schemes forthe barotropic Euler equations (hence, as a particular case, for the shallow waterequations) and for the full Euler equations, based on staggered discretizations. Forstructured meshes, we use the MAC finite volume scheme, and, for general mixedquadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of theRannacher−Turek orCrouzeix−Raviart finiteelements. We first show that a solution to each of these schemes satisfies a discretekinetic energy equation. In the barotropic case, a solution also satisfies a discreteelastic potential balance; integrating these equations over the domain readily yieldsdiscrete counterparts of the stability estimates which are known for the continuousproblem. In the case of the full Euler equations, the scheme relies on the discretizationof the internal energy balance equation, which offers two main advantages: first, we avoidthe space discretization of the total energy, which involves cell-centered andface-centered variables; second, we obtain an algorithm which boils down to a usualpressure correction scheme in the incompressible limit. Consistency (in a weak sense) withthe original total energy conservative equation is obtained thanks to corrective terms inthe internal energy balance, designed to compensate numerical dissipation terms appearingin the discrete kinetic energy inequality. It is then shown in the 1D case, that,supposing the convergence of a sequence of solutions, the limit is an entropy weaksolution of the continuous problem in the barotropic case, and a weak solution in the fullEuler case. Finally, we present numerical results which confirm this theory.

A multiscale spectral generalized finite element method (MS-GFEM) is presented for the solution of large two and three dimensional stress analysis problems inside heterogeneous media. It can be employed to solve problems too large to be solved directly with FE techniques and is designed for implementation on massively parallel machines. The method is multiscale in nature and uses an optimal family of spectrally defined local basis functions over a coarse grid. It is proved that the method has an exponential rate of convergence. To fix ideas we describe its implementation for a two dimensional plane strain problem inside a fiber reinforced composite. Here fibers are separated by a minimum distance however no special assumption on the fiber configuration such as periodicity or ergodicity is made. The implementation of MS-GFEM delivers the discrete solution operator using the same order of operations as the number of fibers inside the computational domain. This implementation is optimal in that the number of operations for solution is of the same order as the input data for the problem. The size of the MS-GFEM matrix used to represent the discrete inverse operator is controlled by the scale of the coarse grid and the convergence rate of the spectral basis and can be of order far less than the number of fibers. This strategy is general and can be applied to the solution of very large FE systems associated with the discrete solution of elliptic PDE.

We present in this paper the formal passage from a kinetic model to the incompressible Navier−Stokes equations for a mixture of monoatomic gases with different masses. The starting point of this derivation is the collection of coupled Boltzmann equations for the mixture of gases. The diffusion coefficients for the concentrations of the species, as well as the ones appearing in the equations for velocity and temperature, are explicitly computed under the Maxwell molecule assumption in terms of the cross sections appearing at the kinetic level.

We propose two new algorithms to improve greedy sampling of high-dimensional functions. While the techniques have a substantial degree of generality, we frame the discussion in the context of methods for empirical interpolation and the development of reduced basis techniques for high-dimensional parametrized functions. The first algorithm, based on a saturation assumption of the error in the greedy algorithm, is shown to result in a significant reduction of the workload over the standard greedy algorithm. In a further improved approach, this is combined with an algorithm in which the train set for the greedy approach is adaptively sparsified and enriched. A safety check step is added at the end of the algorithm to certify the quality of the sampling. Both these techniques are applicable to high-dimensional problems and we shall demonstrate their performance on a number of numerical examples.

We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite generalpartial differential operators. The starting point of our analysis is the DPG methodintroduced by [Demkowicz et al., SIAM J. Numer. Anal.49 (2011) 1788–1809; Zitelli et al., J.Comput. Phys. 230 (2011) 2406–2432]. This discretization resultsin a sparse positive definite linear algebraic system which can be obtained from a saddlepoint problem by an element-wise Schur complement reduction applied to the test space.Here, we show that the abstract framework of saddle point problems and domaindecomposition techniques provide stability and a priori estimates. Toobtain efficient numerical algorithms, we use a second Schur complement reduction appliedto the trial space. This restricts the degrees of freedom to the skeleton. We construct apreconditioner for the skeleton problem, and the efficiency of the discretization and thesolution method is demonstrated by numerical examples.

The instationary Stokes and Navier−Stokes equations are considered in a simultaneously space-timevariational saddle point formulation, so involving both velocities u and pressure p. For the instationaryStokes problem, it is shown that the corresponding operator is a boundedlyinvertible linear mapping between H1 and H'2, both Hilbertspaces H1 and H2 beingCartesian products of (intersections of) Bochner spaces, or duals of those. Based on theseresults, the operator that corresponds to the Navier−Stokes equations is shown to mapH1 into H'2, with a Fréchetderivative that, at any (u,p) ∈H1, is boundedly invertible. These resultsare essential for the numerical solution of the combined pair of velocities and pressureas function of simultaneously space and time. Such a numerical approach allows for theapplication of (adaptive) approximation from tensor products of spatial and temporal trialspaces, with which the instationary problem can be solved at a computational complexitythat is of the order as for a corresponding stationary problem.