Showing papers in "Mathematical Modelling and Numerical Analysis in 2014"

Basic principles of mixed Virtual Element Methods

Abstract: The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of H (div)-conforming vector fields (or, more generally, of (n − 1) − Cochains ). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim of making the basic philosophy clear. However, we consider an arbitrary degree of accuracy k (the Virtual Element analogue of dealing with polynomials of arbitrary order in the Finite Element Framework).

A reduced model for Darcy's problem in networks of fractures

Abstract: Subsurface flows are influenced by the presence of faults and large fractures which act as preferential paths or barriers for the flow. In literature models were proposed to handle fractures in a porous medium as objects of codimension 1. In this work we consider the case of a network of intersecting fractures, with the aim of deriving physically consistent and effective interface conditions to impose at the intersection between fractures. This new model accounts for the angle between fractures at the intersections and allows for jumps of pressure across intersections. This fact permits to describe the flow when fractures are characterized by different properties more accurately with respect to other models that impose pressure continuity. The main mathematical properties of the model, derived in the two-dimensional setting, are analyzed. As concerns the numerical discretization we allow the grids of the fractures to be independent, thus in general non-matching at the intersection, by means of the extended finite element method (XFEM). This increases the flexibility of the method in the case of complex geometries characterized by a high number of fractures.

Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem

Abstract: 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 boundar ...

Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

Abstract: We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough (L^∞) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution H) minimizing the L^2 norm of the source terms; its (pre-)computation involves minimizing O(H^(-d)) quadratic (cell) problems on (super-)localized sub-domains of size O(H ln(1/H)). The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for d ≤ 3, and polyharmonic for d ≥ 4, for the operator -div(a∇.) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method (O(H)) in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincare inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.

Analysis of Compatible Discrete Operator schemes for elliptic problems on polyhedral meshes

Abstract: Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is assembled cellwise from local operators. We analyze two schemes depending on whether the potential degrees of freedom are attached to the vertices or to the cells of the primal mesh. We derive new functional analysis results on the discrete gradient that are the counterpart of the Sobolev embeddings. Then, we identify the two design properties of the local discrete Hodge operators yielding optimal discrete energy error estimates. Additionally, we show how these operators can be built from local nonconforming gradient reconstructions using a dual barycentric mesh. In this case, we also prove an optimal L 2 -error estimate for the potential for smooth solutions. Links with existing schemes (finite elements, finite volumes, mimetic finite differences) are discussed. Numerical results are presented on three-dimensional polyhedral meshes. The detailed material is available from http://hal.archives-ouvertes.fr/hal-00751284

A localized orthogonal decomposition method for semi-linear elliptic problems

Abstract: In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H | log (H ) | where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coefficients, we give a rigorous proof for a linear convergence of the H 1 -error with respect to the coarse mesh size even for rough coefficients. To solve the corresponding system of algebraic equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.

An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations

Abstract: We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the $L^2$-norm in probability of the \mbox{$H^1$-norm} in space of this error scales like $\epsilon$, where $\epsilon$ is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Green's function by Marahrens and the third author.

A modified quasi-boundary value method for the backward time-fractional diffusion problem

Abstract: In this paper, we consider a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine the initial data from a noisy final data. Based on a series expression of the solution, a conditional stability for the initial data is given. Further, we propose a modified quasi-boundary value regularization method to deal with the backward problem and obtain two kinds of convergence rates by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed methods.

On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations

Abstract: In this paper, we propose implicit and semi-implicit in time finite volume schemes for the barotropic Euler equations (hence, as a particular case, for the shallow water equations) and for the full Euler equations, based on staggered discretizations. For structured meshes, we use the MAC finite volume scheme, and, for general mixed quadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of the Rannacher−Turek or Crouzeix−Raviart finite elements. We first show that a solution to each of these schemes satisfies a discrete kinetic energy equation. In the barotropic case, a solution also satisfies a discrete elastic potential balance; integrating these equations over the domain readily yields discrete counterparts of the stability estimates which are known for the continuous problem. In the case of the full Euler equations, the scheme relies on the discretization of the internal energy balance equation, which offers two main advantages: first, we avoid the space discretization of the total energy, which involves cell-centered and face-centered variables; second, we obtain an algorithm which boils down to a usual pressure correction scheme in the incompressible limit. Consistency (in a weak sense) with the original total energy conservative equation is obtained thanks to corrective terms in the internal energy balance, designed to compensate numerical dissipation terms appearing in 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 weak solution of the continuous problem in the barotropic case, and a weak solution in the full Euler case. Finally, we present numerical results which confirm this theory.

Discretized fractional substantial calculus

Abstract: This paper discusses the properties and the numerical discretizations of the fractional substantial integral I-s(v) f(x) = 1/Gamma(v) integral(x)(a) (x-tau)(v-1)e(-sigma(x-tau)) f(tau)d tau, v>0, and the fractional substantial derivative D-s(mu) f(x) = D-s(m) [I-s(v) f(x)], v = m - mu, where D-s = partial derivative/partial derivative x + sigma, sigma can be a constant or a function not related to x, say sigma(y); and m is the smallest integer that exceeds mu. The Fourier transform method and fractional linear multistep method are used to analyze the properties or derive the discretized schemes. And the convergences of the presented discretized schemes with the global truncation error O(h(p)) (p = 1, 2, 3, 4, 5) are theoretically proved and numerically verified.

Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media

Abstract: This paper presents the Vertex Approximate Gradient (VAG) discretization of a two-phase Darcy flow in discrete fracture networks (DFN) taking into account the mass exchange between the matrix and the fracture. We consider the asymptotic model for which the fractures are represented as interfaces of codimension one immersed in the matrix domain with continuous pressures at the matrix fracture interface. Compared with Control Volume Finite Element (CVFE) approaches, the VAG scheme has the advantage to avoid the mixing of the fracture and matrix rocktypes at the interfaces between the matrix and the fractures, while keeping the low cost of a nodal discretization on unstructured meshes. The convergence of the scheme is proved under the assumption that the relative permeabilities are bounded from below by a strictly positive constant but cover the case of discontinuous capillary pressures. The efficiency of our approach compared with CVFE discretizations is shown on a 3D fracture network with very low matrix permeability.

Double greedy algorithms: reduced basis methods for transport dominated problems ∗

Abstract: The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal perfor- mance when compared with the Kolmogorov n-widths of the solution sets. The central ingredient is the construction of computationally feasible "tight" surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.

An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier–Stokes flows

Abstract: We present the current Reduced Basis framework for the efficient numerical approximation of parametrized steady Navier-Stokes equations. We have extended the existing setting developed in the last decade (see e.g. [S. Deparis, SIAM J. Numer. Anal. 46 (2008) 2039-2067; A. Quarteroni and G. Rozza, Numer. Methods Partial Differ. Equ. 23 (2007) 923-948; K. Veroy and A.T. Patera, Int. J. Numer. Methods Fluids 47 (2005) 773-788]) to more general affine and nonaffine parametrizations (such as volume-based techniques), to a simultaneous velocity-pressure error estimates and to a fully decoupled Offline/Online procedure in order to speedup the solution of the reduced-order problem. This is particularly suitable for real-time and many-query contexts, which are both part of our final goal. Furthermore, we present an efficient numerical implementation for treating nonlinear advection terms in a convenient way. A residual-based a posteriori error estimation with respect to a truth, full-order Finite Element approximation is provided for joint pressure/velocity errors, according to the Brezzi-Rappaz-Raviart stability theory. To do this, we take advantage of an extension of the Successive Constraint Method for the estimation of stability factors and of a suitable fixed-point algorithm for the approximation of Sobolev embedding constants. Finally, we present some numerical test cases, in order to show both the approximation properties and the computational efficiency of the derived framework. © 2014 EDP Sciences, SMAI .

High order semi-Lagrangian particle methods for transport equations: numerical analysis and implementation issues

Abstract: This paper is devoted to the definition, analysis and implementation of semi-Lagrangian methods as they result from particle methods combined with remeshing. We give a complete consistency analysis of these methods, based on the regularity and momentum properties of the remeshing kernels, and a stability analysis of a large class of second and fourth order methods. This analysis is supplemented by numerical illustrations. We also describe a general approach to implement these methods in the context of hybrid computing and investigate their performance on GPU processors as a function of their order of accuracy.

A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces

Abstract: Fil: Agnelli, Juan Pablo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Santa Fe. Instituto de Matematica Aplicada "Litoral"; Argentina. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica; Argentina

Multiscale Finite Element approach for “weakly” random problems and related issues

Abstract: We address multiscale elliptic problems with random coefficients that are a perturbation of multiscale deterministic problems. Our approach consists in taking benefit of the perturbative context to suitably modify the classical Finite Element basis into a deterministic multiscale Finite Element basis. The latter essentially shares the same approximation properties as a multiscale Finite Element basis directly generated on the random problem. The specific reference method that we use is the Multiscale Finite Element Method. Using numerical experiments, we demonstrate the efficiency of our approach and the computational speed-up with respect to a more standard approach. In the stationary setting, we provide a complete analysis of the approach, extending that available for the deterministic periodic setting.

A Robust Entropy-Satisfying Finite Volume Scheme for the Isentropic Baer-Nunziato Model

Abstract: We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions

Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation

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

Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy–Forchheimer flow in the fracture

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

Machine computation using the exponentially convergent multiscale spectral generalized finite element method

Abstract: A multiscale spectral generalized nite 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 dened local basis functions over a coarse grid. It is proved that the method has an exponential rate of convergence. To x ideas we describe its implementation for a two dimensional plane strain problem inside a ber reinforced composite. Here bers are separated by a minimum distance however no special assumption on the ber conguration 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 bers 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 bers. This strategy is general and can be applied to the solution of very large FE systems associated with the discrete solution of elliptic PDE.

Error estimates for stokes problem with tresca friction conditions

Abstract: 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 by polymer melt, which can slip on a solid wall. The mixed formulation is based on a dualization of the non-differentiable term which define the slip conditions. Existence and uniqueness of both continuous and discrete solutions of these problems is guaranteed by means of continuous and discrete inf-sup conditions that are proved. Velocity and pressure are approximated by P1 bubble-P1 finite element and piecewise linear elements are used to discretize the Lagrange multiplier associated to the shear stress on the friction boundary. Optimal a priori error estimates are derived using classical tools of finite element analysis and two uncoupled discrete inf-sup conditions for the pressure and the Lagrange multiplier associated to the fluid shear stress.

A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems ∗

Abstract: In this paper, we propose a method for the approximation of the solution of high- dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation for- mats. The method can be seen as a perturbation of a minimal residual method with a measure of the residual corresponding to the error in a specified solution norm. The residual norm can be de- signed such that the resulting low-rank approximations are optimal with respect to particular norms of interest, thus allowing to take into account a particular objective in the definition of reduced order approximations of high-dimensional problems. We introduce and analyze an iterative algorithm that is able to provide an approximation of the optimal approximation of the solution in a given low-rank subset, without any ap rioriinformation on this solution. We also introduce a weak greedy algorithm which uses this perturbed minimal residual method for the computation of successive greedy correc- tions in small tensor subsets. We prove its convergence under some conditions on the parameters of the algorithm. The proposed numerical method is applied to the solution of a stochastic partial differential equation which is discretized using standard Galerkin methods in tensor product spaces. Mathematics Subject Classification. 15A69, 35J50, 41A63, 65D15, 65N12.

A weighted empirical interpolation method: a priori convergence analysis and applications

Abstract: We extend the classical empirical interpolation method to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work. We apply our method to geometric Brownian motion, exponential Karhunen-Loeve expansion and reduced basis approximation of non-ane stochastic elliptic equations. We demonstrate its improved accuracy and eciency over the empirical interpolation method, as well as sparse grid stochastic collocation method.

Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM∗

Abstract: We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a delta-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on delta. We apply the obtained estimates to show exponential convergence with rate O(exp(−b square root N)), N being the number of degrees of freedom and b>0, of a hp-dGFEM discretisation of the Laplace equation based on piecewise harmonic polynomials. This result is an improvement over the classical rate O(exp(−b cubic root N )), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.

A posteriori error estimation for reduced order solutions of parametrized parabolic optimal control problems

Abstract: We consider the ecient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial dierential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving control constraints and that, even in the presence of control constraints, the reduced order optimal control problem and the proposed bounds can be eciently evaluated in an oine-online computational procedure. We also propose two greedy sampling procedures to construct the reduced basis space. Numerical results are presented to conrm the validity of our approach.

Numerical analysis of modular regularization methods for the BDF2 time discretization of the Navier-Stokes equations∗

Abstract: We consider an uncoupled, modular regularization algorithm for approximation of the Navier-Stokes equations. The method is: Step 1.1: Advance the NSE one time step, Step 1.1: Regularize to obtain the approximation at the new time level. Previous analysis of this approach has been for specific time stepping methods in Step 1.1 and simple stabilizations in Step 1.1. In this report we extend the mathematical support for uncoupled, modular stabilization to (i) the more complex and better performing BDF2 time discretization in Step 1.1, and (ii) more general (linear or nonlinear) regularization operators in Step 1.1. We give a complete stability analysis, derive conditions on the Step 1.1 regularization operator for which the combination has good stabilization effects, characterize the numerical dissipation induced by Step 1.1, prove an asymptotic error estimate incorporating the numerical error of the method used in Step 1.1 and the regularizations consistency error in Step 1.1 and provide numerical tests.

Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method

Abstract: The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539–542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the empirical interpolation method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. 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 a three-dimensional acoustic scattering problem solved by a boundary element method.

A hyperbolic model of chemotaxis on a network: a numerical study

Abstract: 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.

Sweeping preconditioners for elastic wave propagation with spectral element methods

Abstract: We present a parallel preconditioning method for the iterative solution of the time-harmonic elastic wave equation which makes use of higher-order spectral elements to reduce pollution error. In particular, the method leverages perfectly matched layer boundary conditions to efficiently approximate the Schur complement matrices of a block LD L T factorization. Both sequential and parallel versions of the algorithm are discussed and results for large-scale problems from exploration geophysics are presented.

Study of a low Mach nuclear core model for two-phase flows with phase transition I: stiffened gas law

Abstract: In this paper, we are interested in modelling the flow of the coolant (water) in a nuclear reactor core. To this end, we use a monodimensional low Mach number model coupled to the stiffened gas law. We take into account potential phase transitions by a single equation of state which describes both pure and mixture phases. In some particular cases, we give analytical steady and/or unsteady solutions which provide qualitative information about the flow. In the second part of the paper, we introduce two variants of a numerical scheme based on the method of characteristics to simulate this model. We study and verify numerically the properties of these schemes. We finally present numerical simulations of a loss of flow accident (LOFA) induced by a coolant pump trip event.