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A modification of the edge stabilization technique is proposed to improve the behavior of the method when solving conservation equations with nonsmooth data or nonsmooth solutions. The key ingredient is tempering the edge stabilization in regions of large gradients through appropriate weights. The new method is shown to preserve the convergence properties of the original method on smooth solutions, and numerical tests indicate that it performs better on nonsmooth solutions. [ABSTRACT FROM AUTHOR]

Abstract: We consider the Crouzeix–Raviart nonconforming finite element method for the Laplace equation. We present four equilibrated flux reconstructions, by direct prescription or by mixed approximation of local Neumann problems, either relying on the original simplicial mesh only or employing a dual mesh. We show that all these reconstructions coincide provided the underlying system of linear algebraic equations is solved exactly. We finally consider an inexact algebraic solve, adjust the flux reconstructions, and point out the differences. [Copyright &y& Elsevier]

We consider two-component (typically, water and hydrogen) compressible liquid-gas porous media flows including mass exchange between phases possibly leading to gas-phase (dis)appearance, as motivated by hydrogen production in underground repositories of radioactive waste. Following recent work by Bourgeat, Jurak, and Smaï, we formulate the governing equations in terms of liquid pressure and dissolved hydrogen density as main unknowns, leading mathematically to a nonlinear elliptic-parabolic system of partial differential equations, in which the equations degenerate when the gas phase disappears. We develop a discontinuous Galerkin method for space discretization, combined with a backward Euler scheme for time discretization and an incomplete Newton method for linearization. Numerical examples deal with gas-phase (dis)appearance, ill-prepared initial conditions, and heterogeneous problem with different rock types. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the stability of a thin liquid film flowing down an inclined wavy plane using a direct numerical solver based on a finite element/arbitrary Lagrangian Eulerian approximation of the free-surface Navier–Stokes equations. We study the dependence of the critical Reynolds number for the onset of surface wave instabilities on the inclination angle, the waviness parameter, and the wavelength parameter, focusing in particular on mild inclinations and relatively large waviness so that the bottom does not fall monotonously. In the present parameter range, shorter wavelengths and higher amplitude for the bottom undulation stabilize the flow. The dependence of the critical Reynolds number evaluated with the Nusselt flow rate on the inclination angle is more complex than the classical relation (5/6 times the cotangent of the inclination angle), but this dependence can be recovered if the actual flow rate at critical conditions is used instead. [Copyright &y& Elsevier]

Abstract: We derive a posteriori error estimates for a class of second-order monotone quasi-linear diffusion-type problems approximated by piecewise affine, continuous finite elements. Our estimates yield a guaranteed and fully computable upper bound on the error measured by the dual norm of the residual, as well as a global error lower bound, up to a generic constant independent of the nonlinear operator. They are thus fully robust with respect to the nonlinearity, thanks to the choice of the error measure. They are also locally efficient, albeit in a different norm, and hence suitable for adaptive mesh refinement. Moreover, they allow to distinguish, estimate separately, and compare the discretization and linearization errors. Hence, the iterative (Newton–Raphson, fixed point) linearization can be stopped whenever the linearization error drops to the level at which it does not affect significantly the overall error. This can lead to important computational savings, as performing an excessive number of unnecessary linearization iterations can be avoided. A strategy combining the linearization stopping criterion and adaptive mesh refinement is proposed and numerically tested for the p-Laplacian. [Copyright &y& Elsevier]

The aim of the present work is to analyze the modified mass method for the dynamic Signorini problem with Coulomb friction. We prove that the space semidiscrete problem is equivalent to an upper semicontinuous one-sided Lipschitz differential inclusion and is, therefore, well-posed. We derive an energy balance. Next, considering an implicit time-integration scheme, we prove that, under a CFL-type condition on the discretization parameters, the fully discrete problem is well-posed. For a fixed discretization in space, we also prove that the fully discrete solutions converge to the space semidiscrete solution when the time step tends to zero. [ABSTRACT FROM AUTHOR]

Abstract: We propose and study a posteriori error estimates for convection–diffusion–reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior-penalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction. We first derive locally computable estimates for the error measured in the energy (semi)norm. These estimates are evaluated using -conforming diffusive and convective flux reconstructions, thereby extending the previous work on pure diffusion problems. The resulting estimates are semi-robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local Péclet and Damköhler numbers. Fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy (semi)norm, a dual norm of the skew-symmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. Numerical experiments are presented to illustrate the theoretical results. [Copyright &y& Elsevier]

A dual-weighted residual error estimation strategy is applied to the modeling error associated with a class of subgrid scale models widely used in turbulent combustion. The approach is implemented and validated for an idealized test problem consisting of a system of multiscale advection-reaction-diffusion equations. The regime of interest corresponds to the bruit of fast reaction. A fully resolved numerical simulation of the resulting very fine reaction scale would be excessively demanding, so typically a subgrid model is used instead to account for its impact at large scales. The class of subgrid models investigated here is based on the asymptotic flamelet approximation, which leads to the so-called flamelet libraries, i.e., precomputed tables expressing the reactive scalars in terms of the appropriate passive scalar. We show that a similar library can be constructed for the dual-based estimator. The performance of the estimation strategy is validated by comparison with exact results for steady and unsteady test cases. Asymptotic scalings are recovered, and new light is shed on some specific implementations of the flamelet models for unsteady test cases. [ABSTRACT FROM AUTHOR]

We propose and analyse a symmetric weighted interior penalty method to approximate in a discontinuous Galerkin framework advection–diffusion equations with anisotropic and discontinuous diffusivity. The originality of the method consists in the use of diffusivity-dependent weighted averages to better cope with locally small diffusivity (or equivalently with locally high Péclet numbers) on fitted meshes. The analysis yields convergence results for the natural energy norm that are optimal with respect to mesh size and robust with respect to diffusivity. The convergence results for the advective derivative are optimal with respect to mesh size and robust for isotropic diffusivity, as well as for anisotropic diffusivity if the cell Péclet numbers evaluated with the largest eigenvalue of the diffusivity tensor are large enough. Numerical results are presented to illustrate the performance of the proposed scheme. [ABSTRACT FROM PUBLISHER]

We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations of advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms: a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfürth on continuous finite elements, that the local lower error bounds can be written with constants involving a cut-off for the ratio of local mesh size to the reciprocal of the square root of the lowest local eignevalue of the diffusion tensor. [ABSTRACT FROM AUTHOR]

We investigate finite element discretizations using functions that are discontinuous in time and continuous in space for European options with local volatility Black–Scholes models. We present an a posteriori error estimate where a user-specified functional of the error is controlled by the inner product of the finite element residual with the solution of a dual problem that involves the density of the target functional as prescribed data. Examples of error functionals are discussed in the context of either option pricing or volatility calibration from market data. The a posteriori error estimator is then localized onto the space-time cells of the computational mesh and implemented in the framework of an adaptive mesh refinement/derefinement algorithm which provides some form of optimal compromise between accuracy requirements and computational costs. Numerical examples illustrate the efficiency of the proposed methodology. [ABSTRACT FROM AUTHOR]

We consider the compressible Navier–Stokes equations for gas flows endowed with general pressure and temperature laws as long as they are compatible with the existence of an entropy and Gibbs relations. We extend the relaxation method introduced for the Euler equations by Coquel and Perthame. Keeping the same “sub-characteristic” conditions for the hyperbolic fluxes and using a consistent splitting of the diffusive fluxes based on a global temperature, we prove the stability of the relaxation system via the sign of the production of a suitable entropy. A first order asymptotic analysis around equilibrium states confirms the stability result. Finally, we present a numerical implementation of the method. To cite this article: E. Bongiovanni et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]

This paper studies the unique continuation problem for the heat equation. We prove a so-called conditional stability estimate for the solution. We are interested in local estimates that are Hölder stable with the weakest possible norms of data on the right-hand side. Such an estimate is useful for the convergence analysis of computational methods dealing with data assimilation. We focus on the case of a known solution at initial time and in some subdomain but that is unknown on the boundary. To the best of our knowledge, this situation has not yet been studied in the literature. [ABSTRACT FROM AUTHOR]

We present a numerical simulation of an axisymmet ric, laminar diffusion flame with finite-rate chemistry on serial and distributed-memory parallel computers. We use the total mass, momentum, energy, and spe cies conservation equations with the compressible Navier-Stokes equations written in vorticity-velocity form. The computational algorithm for solving the re sulting nonlinear coupled elliptic partial differential equations involves damped Newton iterations, Krylov type linear-system solvers, and adaptive mesh refine ment. The results presented here are the first in which a lifted diffusion flame structure is obtained on a non- staggered grid. The numerical solution is in very good agreement with previous numerical and experimental data. [ABSTRACT FROM PUBLISHER]

We design and analyze a new adaptive stabilized finite element method. We construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has inf–sup stability. We formulate this residual minimization as a stable saddle-point problem, which delivers a stabilized discrete solution and a residual representation that drives the adaptive mesh refinement. Numerical results on an advection–reaction model problem show competitive error reduction rates when compared to discontinuous Galerkin methods on uniformly refined meshes and smooth solutions. Moreover, the technique leads to optimal decay rates for adaptive mesh refinement and solutions having sharp layers. • We propose a stabilized FEM based on residual minimization of a discrete-dual norm. • Well-posedness is ensured by standard properties of Discontinuous Galerkin methods. • Mesh adaptivity is efficiently driven by a built-in error estimator of the scheme. • Accuracy and robustness of the scheme is confirmed by 2D and 3D advective problems. • Piecewise discontinuous test spaces, while conforming trial spaces, are considered. [ABSTRACT FROM AUTHOR]

In this work we extend our recently proposed adaptive refinement strategy for h p -finite element approximations of elliptic problems by taking into account an inexact algebraic solver. Namely, on each level of refinement and on each iteration of an (arbitrary) iterative algebraic solver, we compute guaranteed a posteriori error bounds on the algebraic and the total errors in energy norm. The algebraic error is the difference between the inexact discrete solution obtained by an iterative algebraic solver and the (unavailable) exact discrete solution. On the other hand, the total error stands for the difference between the inexact discrete solution and the (unavailable) exact solution of the partial differential equation. For the algebraic error upper bound, we crucially exploit the whole nested hierarchy of h p -finite element spaces created by the adaptive algorithm, whereas the remaining parts of the total error upper and lower bounds are computed using the finest space only. These error bounds allow us to formulate adaptive stopping criteria for the algebraic solver ensuring that the algebraic error does not significantly contribute to the total error. Next, we use the total error bound to mark mesh vertices for refinement via Dörfler's bulk-chasing criterion. On patches associated with marked vertices only, we solve two separate primal finite element problems with homogeneous Dirichlet (Neumann) boundary conditions, which serve to decide between h -, p -, or h p -refinement. Altogether, we show that these ingredients lead to a computable guaranteed bound on the ratio of the total errors of the inexact approximations between successive refinements (the error reduction factor), when the stopping criteria are satisfied. Finally, in a series of numerical experiments, we investigate the practicality of the proposed adaptive solver, the accuracy of our bound on the reduction factor, and show that exponential convergence rates are achieved even in the presence of an inexact algebraic solver. • Adaptive h p -refinement strategy taking into account an inexact algebraic solver. • Computable guaranteed bound on the ratio of errors between successive refinements. • Adaptive stopping criteria for the algebraic solver on each level of refinement. • Numerics: exponential convergence rates in presence of an inexact algebraic solver. [ABSTRACT FROM AUTHOR]

We design and analyze an arbitrary-order hybridized discontinuous Galerkin method to approximate the unique continuation problem subject to the Helmholtz equation. The method is analyzed using conditional stability estimates for the continuous problem, leading to error estimates in norms over interior subdomains of the computational domain. The convergence order reflects the Hölder continuity of the conditional stability estimates and the approximation properties of the finite element space for sufficiently smooth solutions. Under a certain convexity condition, the constant in the estimates is independent of the frequency. Moreover, certain weighted averages of the error are shown to converge independently of the stability properties of the continuous problem. Numerical examples illustrate the performances of the method with respect to the degree of ill-posedness of the problem, increasing polynomial order, and perturbations in the data. [ABSTRACT FROM AUTHOR]

This work is devoted to an analytical description of the dynamics of the static/flowing interface in thin dry granular flows. Our starting point is the asymptotic model derived by Bouchut et al. [Comm. Math. Sci.14 (2016) 2101–2126] from a free surface incompressible model with viscoplastic rheology including a Drucker–Prager yield stress. This asymptotic model is based on the thin-layer approximation (the flow is thin in the direction normal to the topography compared to its down-slope extension), but the equations are not depth-averaged. In addition to the velocity, the model includes a free surface at the top of the flow and a free time-dependent static/flowing interface at the bottom. In the present work, we simplify this asymptotic model by decoupling the space coordinates, and keeping only the dependence on time and on the normal space coordinate Z. We introduce a time- and Z-dependent source term, assumed here to be given, which represents the opposite of the net force acting on the flowing material, including gravity, pressure gradient, and internal friction. We prove several properties of the resulting simplified model that has a time- and Z-dependent velocity and a time-dependent static/flowing interface as unknowns. The crucial advantage of this simplified model is that it can provide explicit solutions in the inviscid case, for different shapes of the source term. These explicit inviscid solutions exhibit a rich behaviour and qualitatively reproduce some physical features observed in granular flows. [ABSTRACT FROM AUTHOR]

We derive equilibrated reconstructions of the Darcy velocity and of the total stress tensor for Biot’s poro-elasticity problem. Both reconstructions are obtained from mixed finite element solutions of local Neumann problems posed over patches of elements around mesh vertices. The Darcy velocity is reconstructed using Raviart–Thomas finite elements and the stress tensor using Arnold–Winther finite elements so that the reconstructed stress tensor is symmetric. Both reconstructions have continuous normal component across mesh interfaces. Using these reconstructions, we derive a posteriori error estimators for Biot’s poro-elasticity problem, and we devise an adaptive space–time algorithm driven by these estimators. The algorithm is illustrated on test cases with analytical solution, on the quarter five-spot problem, and on an industrial test case simulating the excavation of two galleries. [ABSTRACT FROM AUTHOR]

We consider model reduction for linear variational inequalities with parameter-dependent constraints. We study the stability of the reduced problem in the context of a dualized formulation of the constraints using Lagrange multipliers. Our main result is an algorithm that guarantees inf-sup stability of the reduced problem. The algorithm is computationally effective since it can be performed in the offline phase even for parameter-dependent constraints. Moreover, we also propose a modification of the Cone Projected Greedy algorithm so as to avoid ill-conditioning issues when manipulating the reduced dual basis. Our results are illustrated numerically on the frictionless Hertz contact problem between two half-disks with parameter-dependent radius and on the membrane obstacle problem with parameter-dependent obstacle geometry. [ABSTRACT FROM AUTHOR]

The objective of this work is to present a conservative coupling method between an inviscid compressible fluid and a deformable structure undergoing large displacements. The coupling method combines a cut-cell Finite Volume method, which is exactly conservative in the fluid, and a symplectic Discrete Element method for the deformable structure. A time semi-implicit approach is used for the computation of momentum and energy transfer between fluid and solid, the transfer being exactly balanced. The coupling method is exactly mass-conservative (up to round-off errors in the geometry of cut-cells) and exhibits numerically a long-time energy-preservation for the coupled system. The coupling method also exhibits consistency properties, such as conservation of uniform movement of both fluid and solid, absence of numerical roughness on a straight boundary, and preservation of a constant fluid state around a wall having tangential deformation velocity. The performance of the method is assessed on test cases involving shocked fluid flows interacting with two and three-dimensional deformable solids undergoing large displacements. [ABSTRACT FROM AUTHOR]

We study low-order reconstruction operators on polyhedral meshes, providing a unified framework for degrees of freedom attached to vertices, edges, faces, and cells. We present two equivalent sets of design properties and draw links with the literature. In particular, the two-level construction based on a P 0 -consistent and a stabilization part provides a systematic way of designing these operators. We present a simple example of piecewise constant reconstruction in each mesh cell, relying on geometric identities to fulfill the design properties on polyhedral meshes. Finally, we use these reconstruction operators to define a Hodge inner product and build Compatible Discrete Operator schemes, and we test the influence of the reconstruction operators in terms of accuracy and computational efficiency on an anisotropic diffusion problem. [ABSTRACT FROM AUTHOR]

We design an unfitted hybrid high-order (HHO) method for the wave equation. The wave propagates in a domain where a curved interface separates subdomains with different material properties. The key feature of the discretization method is that the interface can cut more or less arbitrarily through the mesh cells. We address both the second-order formulation in time of the wave equation and its reformulation as a first-order system. We prove H 1 -error estimates for a space semi-discretization in space of the second-order formulation, leading to optimal convergence rates for smooth solutions. Numerical experiments illustrate the theoretical findings and show that the proposed numerical schemes can be used to simulate accurately the propagation of acoustic waves in heterogeneous media, with meshes that are not fitted to the geometry. For the second-order formulation, the implicit, second-order accurate Newmark scheme is used for the time discretization, whereas (diagonally-implicit or explicit) Runge–Kutta schemes up to fourth-order accuracy are used for the first-order formulation. In the explicit case, we study the CFL condition on the time step and observe that the unfitted approach combined with local cell agglomeration leads to a comparable condition as when using fitted meshes. [ABSTRACT FROM AUTHOR]

We design and analyze a Hybrid High-Order (HHO) method on unfitted meshes to approximate elliptic interface problems by means of a consistent penalty method à la Nitsche. The curved interface can cut through the mesh cells in a rather general fashion. Robustness with respect to the cuts is achieved by using a cell agglomeration technique, and robustness with respect to the contrast in the diffusion coefficients is achieved by using a different gradient reconstruction on each side of the interface. A key novel feature of the gradient reconstruction is to incorporate a jump term across the interface, thereby releasing the Nitsche penalty parameter from the constraint of being large enough. Error estimates with optimal convergence rates are established. A robust cell agglomeration procedure limiting the agglomerations to the nearest neighbors is devised. Numerical simulations for various interface shapes corroborate the theoretical results. [ABSTRACT FROM AUTHOR]