For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P1-finite element discretisation of Cauchy–Born nonlinear elasticity, this paper adresses the question whether patch test consistency (or, absence of ghost forces) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction, (iii) volumetric scaling of the interface correction, and (iv) connectedness of the atomistic region. The extent to which these assumptions are necessary is discussed in detail.

The numerical solution of the Hartree-Fock equations is a central problem in quantumchemistry for which numerous algorithms exist. Attempts to justify these algorithmsmathematically have been made, notably in [E. Cancès and C. Le Bris, Math. Mod.Numer. Anal. 34 (2000) 749–774], but, to our knowledge, nocomplete convergence proof has been published, except for the large-Zresult of [M. Griesemer and F. Hantsch, Arch. Rational Mech. Anal. (2011)170]. In this paper, we prove the convergence of a natural gradient algorithm, using agradient inequality for analytic functionals due to Łojasiewicz [Ensemblessemi-analytiques. Institut des Hautes Études Scientifiques (1965)]. Then,expanding upon the analysis of [E. Cancès and C. Le Bris, Math. Mod. Numer. Anal.34 (2000) 749–774], we prove convergence results for the Roothaanand Level-Shifting algorithms. In each case, our method of proof provides estimates on theconvergence rate. We compare these with numerical results for the algorithms studied.

In the framework of an explicitly correlated formulation of the electronic Schrödingerequation known as the transcorrelated method, this work addresses some fundamental issuesconcerning the feasibility of eigenfunction approximation by hyperbolic wavelet bases.Focusing on the two-electron case, the integrability of mixed weak derivatives ofeigenfunctions of the modified problem and the improvement compared to the standardformulation are discussed. Elements of a discretization of the eigenvalue problem based onorthogonal wavelets are described, and possible choices of tensor product bases arecompared especially from an algorithmic point of view. The use of separable approximationsof potential terms for applying operators efficiently is studied in detail, and estimatesfor the error due to this further approximation are given.

Some electromagnetic materials have, in a given frequency range, an effective dielectric permittivity and/or a magnetic permeability which are real-valued negative coefficients when dissipation is neglected. They are usually called metamaterials. We study a scalar transmission problem between a classical dielectric material and a metamaterial, set in an open, bounded subset of Rd, with d = 2,3. Our aim is to characterize occurences where the problem is well-posed within the Fredholm (or coercive + compact) framework. For that, we build some criteria, based on the geometry of the interface between the dielectric and the metamaterial. The proofs combine simple geometrical arguments with the approach of T-coercivity, introduced by the first and third authors and co-worker. Furthermore, the use of localization techniques allows us to derive well-posedness under conditions that involve the knowledge of the coefficients only near the interface. When the coefficients are piecewise constant, we establish the optimality of the criteria.

We consider linear elliptic problems with variable coefficients, which may sharply change values and have a complex behavior in the domain. For these problems, a new combined discretization-modeling strategy is suggested and studied. It uses a sequence of simplified models, approximating the original one with increasing accuracy. Boundary value problems generated by these simplified models are solved numerically, and the approximation and modeling errors are estimated by a posteriori estimates of functional type. An efficient numerical strategy is based upon balancing the modeling and discretization errors, which provides an economical way of finding an approximate solution with an a priori given accuracy. Numerical tests demonstrate the reliability and efficiency of this combined modeling-discretization method.

This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory problem. This leads to methods which, counter-intuitively to those developed according to standard numerical reasoning, exhibit improved performance with growing frequency of oscillation.

Since matrix compression has paved the way for discretizing the boundary integralequation formulations of electromagnetics scattering on very fine meshes, preconditionersfor the resulting linear systems have become key to efficient simulations. Operatorpreconditioning based on Calderón identities has proved to be a powerful device fordevising preconditioners. However, this is not possible for the usual first-kind boundaryformulations for electromagnetic scattering at general penetrable composite obstacles. Wepropose a new first-kind boundary integral equation formulation following the reasoningemployed in [X. Clayes and R. Hiptmair, Report 2011-45, SAM, ETH Zürich (2011)] foracoustic scattering. We call it multi-trace formulation, because itsunknowns are two pairs of traces on interfaces in the interior of the scatterer. We give acomprehensive analysis culminating in a proof of coercivity, and uniqueness and existenceof solution. We establish a Calderón identity for the multi-trace formulation, which formsthe foundation for operator preconditioning in the case of conforming Galerkin boundaryelement discretization.

We present a heterogeneous finite element method for the solution of a high-dimensionalpopulation balance equation, which depends both the physical and the internal propertycoordinates. The proposed scheme tackles the two main difficulties in the finite elementsolution of population balance equation: (i) spatial discretization with the standardfinite elements, when the dimension of the equation is more than three, (ii) spuriousoscillations in the solution induced by standard Galerkin approximation due to pureadvection in the internal property coordinates. The key idea is to split thehigh-dimensional population balance equation into two low-dimensional equations, anddiscretize the low-dimensional equations separately. In the proposed splitting scheme, theshape of the physical domain can be arbitrary, and different discretizations can beapplied to the low-dimensional equations. In particular, we discretize the physical andinternal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG)finite elements, respectively. The stability and error estimates of the Galerkin/SUPGfinite element discretization of the population balance equation are derived. It is shownthat a slightly more regularity, i.e.the mixed partial derivatives of the solution has to be bounded, is necessary for theoptimal order of convergence. Numerical results are presented to support the analysis.

In this article we study discontinuous Galerkin finite element discretizations of linearsecond-order elliptic partial differential equations with Dirac delta right-hand side. Inparticular, assuming that the underlying computational mesh is quasi-uniform, we derive ana priori bound on the error measured in terms of theL2-norm. Additionally, we develop residual-based aposteriori error estimators that can be used within an adaptive mesh refinementframework. Numerical examples for the symmetric interior penalty scheme are presentedwhich confirm the theoretical results.

We derive and analyze adaptive solvers for boundary value problems in which thedifferential operator depends affinely on a sequence of parameters. These methods convergeuniformly in the parameters and provide an upper bound for the maximal error. Numericalcomputations indicate that they are more efficient than similar methods that control theerror in a mean square sense.

This paper is concerned with the numerical approximation of mean curvature flow t → Ω(t) satisfying an additional inclusion-exclusion constraint Ω1 ⊂ Ω(t) ⊂ Ω2. Classical phase field model to approximate these evolving interfaces consists in solving the Allen-Cahn equation with Dirichlet boundary conditions. In this work, we introduce a new phase field model, which can be viewed as an Allen Cahn equation with a penalized double well potential. We first justify this method by a Γ-convergence result and then show some numerical comparisons of these two different models.

We develop a Discrete Element Method (DEM) for elastodynamics using polyhedral elements. We show that for a given choice of forces and torques, we recover the equations of linear elastodynamics in small deformations. Furthermore, the torques and forces derive from a potential energy, and thus the global equation is an Hamiltonian dynamics. The use of an explicit symplectic time integration scheme allows us to recover conservation of energy, and thus stability over long time simulations. These theoretical results are illustrated by numerical simulations of test cases involving large displacements.

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.