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The Reduced Basis Method can be exploited in an efficient way only if the so-called affine dependence assumption on the operator and right-hand side of the considered problem with respect to the parameters is satisfied. When it is not, the Empirical Interpolation Method is usually used to recover this assumption approximately. In both cases, the Reduced Basis Method requires to access and modify the assembly routines of the corresponding computational code, leading to an intrusive procedure. In this work, we derive variants of the EIM algorithm and explain how they can be used to turn the Reduced Basis Method into a nonintrusive procedure. We present examples of aeroacoustic problems solved by integral equations and show how our algorithms can benefit from the linear algebra tools available in the considered code. [ABSTRACT FROM AUTHOR]

Compatible discrete operator schemes preserve basic properties of the continuous model at the discrete level. They combine discrete differential operators that discretize exactly topological laws and discrete Hodge operators that approximate constitutive relations. We devise and analyse two families of such schemes for the Stokes equations in curl formulation, with the pressure degrees of freedom located at either mesh vertices or cells. The schemes ensure local mass and momentum conservation. We prove discrete stability by establishing novel discrete Poincaré inequalities. Using commutators related to the consistency error, we derive error estimates with first-order convergence rates for smooth solutions. We analyse two strategies for discretizing the external load, so as to deliver tight error estimates when the external load has a large curl-free or divergence-free part. Finally, numerical results are presented on three-dimensional polyhedral meshes. [ABSTRACT FROM AUTHOR]

We design and analyze an approximation method for advection-diffusion-reaction equations where the (generalized) degrees of freedom are polynomials of order k ≥ 0 at mesh faces. The method hinges on local discrete reconstruction operators for the diffusive and advective derivatives and a weak enforcement of boundary conditions. Fairly general meshes with polytopal and nonmatching cells are supported. Arbitrary polynomial orders can be considered, including the case k = 0, which is closely related to mimetic finite difference/mixed-hybrid finite volume methods. The error analysis covers the full range of Peclet numbers, including the delicate case of local degeneracy where diffusion vanishes on a strict subset of the domain. Computational costs remain moderate since the use of face unknowns leads to a compact stencil with reduced communications. Numerical results are presented. [ABSTRACT FROM AUTHOR]

We show how to recover equilibrated face tractions for the Hybrid High-Order method for linear elasticity recently introduced in [1] , and prove that these tractions are optimally convergent. [ABSTRACT FROM AUTHOR]

We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by the mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme. [ABSTRACT FROM AUTHOR]

We extend the Hybrid High-Order method introduced by the authors for the Poisson problem to problems with heterogeneous/anisotropic diffusion. The cornerstone is a local discrete gradient reconstruction from element- and face-based polynomial degrees of freedom. Optimal error estimates are proved. [ABSTRACT FROM AUTHOR]

We devise an arbitrary-order locking-free method for linear elasticity. The method relies on a pure-displacement (primal) formulation and leads to a symmetric, positive definite system matrix with compact stencil. The degrees of freedom are vector-valued polynomials of arbitrary order k ⩾ 1 on the mesh faces, so that in three space dimensions, the lowest-order scheme only requires 9 degrees of freedom per mesh face. The method can be deployed on general polyhedral meshes. The key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems. The discrete problem is assembled cell-wise using these operators and a high-order stabilization bilinear form. Locking-free error estimates are derived for the energy norm and for the L 2 -norm of the displacement, with optimal convergence rates of order ( k + 1 ) and ( k + 2 ) , respectively, for smooth solutions on general meshes. The theoretical results are confirmed numerically, and the CPU cost is evaluated on both standard and polygonal meshes. [ABSTRACT FROM AUTHOR]

We study the double discrete Legendre-Fenchel transform (LFT) to approximate the convex envelope of a given function. We analyze the convergence of the double discrete LFT in the multivariate case based on previous convergence results for the discrete LFT. We focus our attention on the grid on which the second discrete LFT is computed (dual grid); its choice has great impact on the accuracy of the resulting approximation of the convex envelope. Then, we present an improvement (both in time and accuracy) to the standard algorithm based on a change in the factorization order for the second discrete LFT. This modification is particularly beneficial for bivariate functions. Moreover, we introduce a method for handling functions that are unbounded outside sets of general shape. We also present some situations in which the selection of the dual grid is crucial, and show that it is possible to choose a dual grid of arbitrary size without increasing the memory requirements of the algorithm. Finally, we apply our algorithm to the study of phase separation in non-ideal ionic solutions. [ABSTRACT FROM AUTHOR]

We are interested in solving the unique continuation problem for the heat equation, i.e., we want to reconstruct the solution of the heat equation in a target space-time subdomain given its (noised) value in a subset of the computational domain. Both initial and boundary data can be unknown. We discretize this problem using a space-time discontinuous Galerkin method (including hybrid variables in space) and look for the solution that minimizes a discrete Lagrangian. We establish discrete inf-sup stability and bound the consistency error, leading to a priori estimates on the residual. Owing to the ill-posed nature of the problem, an additional estimate on the residual dual norm is needed to prove the convergence of the discrete solution to the exact solution in the energy norm in the target space-time subdomain. This is achieved by combining the above results with a conditional stability estimate at the continuous level. The rate of convergence depends on the conditional stability, the approximation order in space and in time, and the size of the perturbations in data. Quite importantly, the weight of the regularization term depends on the time step and the mesh size, and we show how to choose it to preserve the best possible decay rates on the error. Finally, we run numerical simulations to assess the performance of the method in practice. [ABSTRACT FROM AUTHOR]

Abstract: We consider the convected Helmholtz equation modeling linear acoustic propagation at a fixed frequency in a subsonic flow around a scattering object. The flow is supposed to be uniform in the exterior domain far from the object, and potential in the interior domain close to the object. Our key idea is the reformulation of the original problem using the Prandtl–Glauert transformation on the whole flow domain, yielding (i) the classical Helmholtz equation in the exterior domain and (ii) an anisotropic diffusive PDE with skew-symmetric first-order perturbation in the interior domain such that its transmission condition at the coupling boundary naturally fits the Neumann condition from the classical Helmholtz equation. Then, efficient off-the-shelf tools can be used to perform the BEM–FEM coupling, leading to two novel variational formulations for the convected Helmholtz equation. The first formulation involves one surface unknown and can be affected by resonant frequencies, while the second formulation avoids resonant frequencies and involves two surface unknowns. Numerical simulations are presented to compare the two formulations. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]