We prove the discrete compactness property of the edge elements of any order on a classof anisotropically refined meshes on polyhedral domains. The meshes, made up oftetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl.Sci. 21 (1998) 519–549]. They are appropriately graded nearsingular corners and edges of the polyhedron.

We prove error estimates for the ultra weak variational formulation (UWVF) in 3D linearelasticity. We show that the UWVF of Navier’s equation can be derived as an upwinddiscontinuous Galerkin method. Using this observation, error estimates are investigatedapplying techniques from the theory of discontinuous Galerkin methods. In particular, wederive a basic error estimate for the UWVF in a discontinuous Galerkin type norm and thenan error estimate in the L2(Ω) norm in terms of the bestapproximation error. Our final result is an L2(Ω) norm errorestimate using approximation properties of plane waves to give an estimate for the orderof convergence. Numerical examples are presented.

In earlier work we have studied a method for discretization in time of a parabolic problem, which consists of representing the exact solution as an integral in the complex plane and then applying a quadrature formula to this integral. In application to a spatially semidiscrete finite-element version of the parabolic problem, at each quadrature point one then needs to solve a linear algebraic system having a positive-definite matrix with a complex shift. We study iterative methods for such systems, considering the basic and preconditioned versions of first the Richardson algorithm and then a conjugate gradient method.

A new approach for computationally efficient estimation of stability factors forparametric partial differential equations is presented. The general parametric bilinearform of the problem is approximated by two affinely parametrized bilinear forms atdifferent levels of accuracy (after an empirical interpolation procedure). The successiveconstraint method is applied on the coarse level to obtain a lower bound for the stabilityfactors, and this bound is extended to the fine level by adding a proper correction term.Because the approximate problems are affine, an efficient offline/online computationalscheme can be developed for the certified solution (error bounds and stability factors) ofthe parametric equations considered. We experiment with different correction terms suitedfor a posteriori error estimation of the reduced basis solution ofelliptic coercive and noncoercive problems.

From Kantorovich’s theory we present a semilocal convergence result for Newton’s methodwhich is based mainly on a modification of the condition required to the second derivativeof the operator involved. In particular, instead of requiring that the second derivativeis bounded, we demand that it is centered. As a consequence, we obtain a modification ofthe starting points for Newton’s method. We illustrate this study with applications tononlinear integral equations of mixed Hammerstein type.

The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.

We consider the Laplace equation posed in a three-dimensional axisymmetric domain. Wereduce the original problem by a Fourier expansion in the angular variable to a countablefamily of two-dimensional problems. We decompose the meridian domain, assumed polygonal,in a finite number of rectangles and we discretize by a spectral method. Then we describethe main features of the mortar method and use the algorithm Strang Fix to improve theaccuracy of our discretization.

The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-typeapproximate Riemann solver for a hyperbolic nonconservative PDE system arising in aturbidity current model. The main difficulties come from the nonconservative nature of thesystem. A general strategy to derive simple approximate Riemann solvers fornonconservative systems is introduced, which is applied to the turbidity current model toobtain two different HLLC solvers. Some results concerning the non-negativity preservingproperty of the corresponding numerical methods are presented. The numerical resultsprovided by the two HLLC solvers are compared between them and also with those obtainedwith a Roe-type method in a number of 1d and 2d test problems. This comparison shows that,while the quality of the numerical solutions is comparable, the computational cost of theHLLC solvers is lower, as only some partial information of the eigenstructure of thematrix system is needed.

For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal points or those edges/nodal points where the support of the corresponding basis function has changed during the refinement process. The adaptive mesh refinement is based on Dörfler marking for residual-type a posteriori error estimators and the newest vertex bisection strategy. Using the abstract Schwarz theory of multilevel iterative schemes, quasi-optimal convergence of the LMM is shown, i.e., the convergence rates are independent of mesh sizes and mesh levels provided the coarsest mesh is chosen sufficiently fine. The theoretical findings are illustrated by the results of some numerical examples.

In this paper we analyze the consistency, the accuracy and some entropy properties ofparticle methods with remeshing in the case of a scalar one-dimensional conservation law.As in [G.-H. Cottet and L. Weynans, C. R. Acad. Sci. Paris, Ser. I343 (2006) 51–56] we re-write particle methods with remeshing inthe finite-difference formalism. This allows us to prove the consistency of these methods,and accuracy properties related to the accuracy of interpolation kernels. Cottet and Magnidevised recently in [G.-H. Cottet and A. Magni, C. R. Acad. Sci. Paris, Ser. I347 (2009) 1367–1372] and [A. Magni and G.-H. Cottet, J.Comput. Phys. 231 (2012) 152–172] TVD remeshing schemes forparticle methods. We extend these results to the nonlinear case with arbitrary velocitysign. We present numerical results obtained with these new TVD particle methods for theEuler equations in the case of the Sod shock tube. Then we prove that with these new TVDremeshing schemes the particle methods converge toward the entropy solution of the scalarconservation law.

Searching for the optimal partitioning of a domain leads to the use of the adjoint methodin topological asymptotic expansions to know the influence of a domain perturbation on acost function. Our approach works by restricting to local subproblems containing theperturbation and outperforms the adjoint method by providing approximations of higherorder. It is a universal tool, easily adapted to different kinds of real problems and doesnot need the fundamental solution of the problem; furthermore our approach allows toconsider finite perturbations and not infinitesimal ones. This paper provides theoreticaljustifications in the linear case and presents some applications with topologicalperturbations, continuous perturbations and mesh perturbations. This proposed approach canalso be used to update the solution of singularly perturbed problems.