We consider an incompressible flow problem in a N-dimensional fracturedporous domain (Darcy’s problem). The fracture is represented by a(N − 1)-dimensional interface, exchanging fluid with the surroundingmedia. In this paper we consider the lowest-order(ℝ T0, ℙ0) Raviart-Thomas mixed finite elementmethod for the approximation of the coupled Darcy’s flows in the porous media and withinthe fracture, with independent meshes for the respective domains. This is achieved thanksto an enrichment with discontinuous basis functions on triangles crossed by the fractureand a weak imposition of interface conditions. First, we study the stability andconvergence properties of the resulting numerical scheme in the uncoupled case, when theknown solution of the fracture problem provides an immersed boundary condition. We detailthe implementation issues and discuss the algebraic properties of the associated linearsystem. Next, we focus on the coupled problem and propose an iterative porousdomain/fracture domain iterative method to solve for fluid flow in both the porous mediaand the fracture and compare the results with those of a traditional monolithic approach.Numerical results are provided confirming convergence rates and algebraic propertiespredicted by the theory. In particular, we discuss preconditioning and equilibrationtechniques to make the condition number of the discrete problem independent of theposition of the immersed interface. Finally, two and three dimensional simulations ofDarcy’s flow in different configurations (highly and poorly permeable fracture) areanalyzed and discussed.

Low-order nonconforming Galerkin methods will be analyzed for second-order elliptic equations subjected to Robin, Dirichlet, or Neumann boundary conditions. Both simplicial and rectangular elements will be considered in two and three dimensions. The simplicial elements will be based on P 1, as for conforming elements; however, it is necessary to introduce new elements in the rectangular case. Optimal order error estimates are demonstrated in all cases with respect to a broken norm in H 1(Ω) and in the Neumann and Robin cases in L 2(Ω).

The discretisation of the Oseen problem by finite element methods may sufferin general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi)condition can be violated. Second, spurious oscillationsoccur due to the dominating convection. One way to overcome bothdifficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we show that the fulfilment of a local inf-sup condition between approximation andprojection spaces allows to construct an interpolation with additional orthogonality properties. Based on this special interpolation, optimal a-priori error estimates are shown with error constants independent of the Reynolds number. Applying the general theory,we extend the results of Braack and Burman for the standard two-level versionof the local projection stabilisation to discretisations of arbitrary order onsimplices, quadrilaterals, and hexahedra. Moreover, our general theory allowsto derive a novel class of local projection stabilisation by enrichment of the approximation spaces. This class of stabilised schemes uses approximation and projection spaces defined on the same mesh and leads tomuch more compact stencils than in the two-level approach. Finally, on simplices, the spectral equivalence of the stabilising terms of the local projection method and the subgrid modelling introduced by Guermond is shown. This clarifies the relation of the local projection stabilisation to the variational multiscaleapproach.

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 study the asymptotic behaviour of the following nonlinear problem: $$\{\begin{array}{ll}-{\rm div}(a( Du_h))+\vert u_h\vert^{p-2}u_h =f \quad\hbox{in }\Omega_h, a( Du_h)\cdot\nu = 0 \quad\hbox{on }\partial\Omega_h, \end{array}.$$

in a domain Ωh of $\mathbb{R}^n$ whose boundary ∂Ωh contains an oscillating part with respect to hwhen h tends to ∞. The oscillating boundary is defined by a set of cylinders with axis 0xn that are h -1-periodically distributed. We prove that the limit problem in the domain corresponding tothe oscillating boundary identifieswith a diffusion operator with respect tox n coupled with an algebraic problemfor the limit fluxes.

We propose a simple numerical method for capturing thesteady state solution of hyperbolic systems with geometricalsource terms. We usethe interface value, rather than the cell-averages, for the source terms that balance the nonlinear convectionat the cell interface, allowing the numerical capturing of the steadystate with a formal high order accuracy. This method applies to Godunovor Roe type upwind methods butrequires no modification of the Riemann solver. Numerical experiments on scalar conservationlaws and the one dimensional shallow water equationsshow much better resolution of the steady state than the conventionalmethod, with almost no new numerical complexity.

We consider the development and analysis of local discontinuous Galerkin methods forfractional diffusion problems in one space dimension, characterized by having fractionalderivatives, parameterized by β ∈[1, 2]. After demonstrating that aclassic approach fails to deliver optimal order of convergence, we introduce a modifiedlocal numerical flux which exhibits optimal order of convergence 𝒪(hk + 1) uniformly across the continuous range between pureadvection (β = 1) and pure diffusion (β = 2). In the twoclassic limits, known schemes are recovered. We discuss stability and present an erroranalysis for the space semi-discretized scheme, which is supported through a fewexamples.

The goal of this paper is to construct a first-order upwind schemefor solving the system of partial differential equations governing theone-dimensional flow of two superposed immiscible layers of shallow waterfluids. This is done by generalizing a numerical scheme presented byBermúdez and Vázquez-Cendón [3, 6, 27] for solving one-layer shallow water equations, consisting in a Q-scheme with a suitable treatment of the source terms.The difficulty in the two layer system comes from the coupling terms involving some derivatives of the unknowns.Due to these terms, a numerical scheme obtained by performing the upwinding of each layer, independently from the other one, can be unconditionally unstable.In order to define a suitable numerical scheme with global upwinding,we first consider an abstract system that generalizes the problem under study.This system is not a system of conservation laws but, nevertheless,Roe's method can be applied to obtain an upwind scheme based on Approximate Riemann State Solvers. Following this, we present some numerical tests to validate the resulting schemes and to highlight the fact that, in general, numerical schemes obtained by applying a Q-scheme to each separate conservation law of the system do not yield good results. First, a simple system of coupled Burgers' equations is considered. Then, the Q-scheme obtained is applied to the two-layer shallow water system.

We are concerned with a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in medium-frequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. We focus on a class of Trefftz-type discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give a priori convergence estimates for the h-version of these plane wave discontinuous Galerkin methods in two dimensions. To that end, we develop new inverse and approximation estimates for plane waves and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion.

We consider space semi-discretizations of the 1-d wave equation in a boundedinterval with homogeneous Dirichlet boundary conditions. We analyze the problemof boundary observability, i.e., the problem of whether the total energy ofsolutions can be estimated uniformly in terms of the energy concentrated on theboundary as the net-spacing h → 0. We prove that, due to the spurious modesthat the numerical scheme introduces at high frequencies, there is no such auniform bound. We prove however a uniform bound in a subspace of solutionsgenerated by the low frequencies of the discrete system. When h → 0 thisfinite-dimensional spaces increase and eventually cover the whole space. Wethus recover the well-known observability property of the continuous systemas the limit of discrete observability estimates as the mesh size tends tozero.We consider both finite-difference and finite-element semi-discretizations.

Proper orthogonal decomposition (POD) is apowerful technique for model reduction of non-linear systems. Itis based on a Galerkin type discretization with basis elementscreated from the dynamical system itself. In the context ofoptimal control this approach may suffer from the fact that thebasis elements are computed from a reference trajectory containingfeatures which are quite different from those of the optimallycontrolled trajectory. A method is proposed which avoids thisproblem of unmodelled dynamics in the proper orthogonaldecomposition approach to optimal control. It is referred to asoptimality system proper orthogonal decomposition (OS-POD).

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 study some discretization schemes for diffusive flows in heterogeneous anisotropic porous media. We first introduce the notion of gradient scheme, and show that several existing schemes fall into this framework. Then, we construct two new gradient schemes which have the advantage of a small stencil. Numerical results obtained for real reservoir meshes show the efficiency of the new schemes, compared to existing ones.

In dimension one it is proved that the solution to a total variation-regularizedleast-squares problem is always a function which is "constant almost everywhere" ,provided that the data are in a certain sense outside the range of the operatorto be inverted. A similar, but weaker result is derived in dimension two.

We study a depth-averaged model of gravity-driven flows made ofsolid grains and fluid, moving over variable basal surface. In particular, we are interested in applicationsto geophysical flows such as avalanches and debris flows,which typically contain both solid material and interstitial fluid.The model system consists of mass and momentum balance equations for thesolid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the unknowns,and by interphase drag source terms. The system is hyperbolic at leastwhen the difference between solid and fluid velocities is sufficiently small.We solve numerically the one-dimensional model equations by a high-resolution finite volume scheme based on a Roe-type Riemann solver. Well-balancing oftopography source terms is obtained via a technique that includesthese contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problemsof perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions.

The incompressible MHD equations couple Navier-Stokes equations with Maxwell's equationsto describe the flow of a viscous, incompressible, and electrically conducting fluid ina Lipschitz domain $\Omega \subset \mathbb{R}^3$ .We verify convergence of iterates of different coupling anddecoupling fully discrete schemes towards weak solutions forvanishing discretization parameters. Optimal first order of convergence is shown in the presence of strong solutions for a splitting scheme which decouplesthe computation of velocity field, pressure, and magnetic fields atevery iteration step.

The main goal of this article is to establish a priori and a posteriori error estimates for the numerical approximation of some non linear ellipticproblems arising in glaciology. The stationary motion of a glacier is givenby a non-Newtonian fluid flow model which becomes, in a firsttwo-dimensional approximation, the so-called infinite parallel sided slabmodel. The approximation of this model is made by a finite element methodwith piecewise polynomial functions of degree 1. Numerical results show thatthe theoretical results we have obtained are almost optimal.

We semi-discretize in space a time-dependent Navier-Stokes systemon a three-dimensional polyhedron by finite-elements schemesdefined on two grids. In the first step, the fully non-linearproblem is semi-discretized on a coarse grid, with mesh-size H.In the second step, the problem is linearized by substitutinginto the non-linear term, the velocity u H computed at stepone, and the linearized problem is semi-discretized on a finegrid with mesh-size h. This approach is motivated by the factthat, on a convex polyhedron and under adequate assumptions on thedata, the contribution of u H to the error analysis ismeasured in the L 2 norm in space and time, and thus, for thelowest-degree elements, is of the order of H 2. Hence, an errorof the order of h can be recovered at the second step, providedh = H 2.

Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference schemes in the stationary case with constant coefficients and in the time-dependent case with variable coefficients by using control theory and probabilistic methods. In this paper we are able to handle variable coefficients by a purely analytical method. In our opinion this way is far simpler and, for the cases we can treat, it yields a better rate of convergence than Krylov obtains in the variable coefficients case.

In this article, we investigate numerical schemes for solvinga three component Cahn-Hilliard model. The space discretization isperformed by usinga Galerkin formulation and the finite element method.Concerning the time discretization,the main difficulty is to write a scheme ensuring,at the discrete level, the decrease of the free energyand thus the stability of the method.We study three different schemes and proveexistence and convergence theorems. Theoretical results areillustrated by various numerical examples showing that the new semi-implicitdiscretization that we propose seems to be a good compromise between robustnessand accuracy.