The gradient discretisation method for linear advection problems

We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped P1 conforming and non conforming finite element and on the hybrid finite volume method

A unified analysis of elliptic problems with various boundary conditions and their approximation

We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue--Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods. The considered approximations can be conforming, or not (that is, the approximation functions can belong to the energy space of the problem, or not), and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to other models, including flows in fractured medium, elasticity equations and diffusion equations on manifolds. A by-product of the analysis is an apparently novel result on the equivalence between general Poincar{\'e} inequalities and the surjectivity of the divergence operator in appropriate spaces

A comparative study between a brain Na+,K+-ATPase inhibitor (endobain E) and ascorbic acid

In the search of Na+,K+-ATPase modulators, we have reported the isolation by gel filtration and HPLC of a brain fraction, termed endobain E, which highly inhibits Na+,K+-ATPase activity. In the present study we compared some properties of endobain E with those of ascorbic acid. Kinetic experiments assaying synaptosomal membrane K+-p-nitrophenylphosphatase (K+p-NPPase) activity in the presence of endobain E or ascorbic acid showed that in neither case did enzyme inhibition prove competitive in nature versus K+ or p-NPP concentration. At pH 5.0, endobain E and ascorbic acid maximal UV absorbance was 266 and 258 nm, respectively; alkalinization to pH 14.0 led to absorption drop and shift for endobain E but to absorbance disappearance for ascorbic acid. After cysteine treatment, endobain E absorbance decreased, whereas that of ascorbic acid remained unaltered; iodine treatment led to absorbance drop and shift for endobain E but to absorbance disappearance for ascorbic acid. HPLC analysis of endobain E disclosed the presence of two components: one eluting with retention time and UV spectrum indistinguishable from those of ascorbic acid and a second, as yet unidentified, both exerting Na+,K+-ATPase inhibition.Fil: Rodriguez, Georgina Emma. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Biología Celular y Neurociencia "Prof. Eduardo de Robertis". Universidad de Buenos Aires. Facultad de Medicina. Instituto de Biología Celular y Neurociencia; ArgentinaFil: Herbin, T.. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Biología Celular y Neurociencia "Prof. Eduardo de Robertis". Universidad de Buenos Aires. Facultad de Medicina. Instituto de Biología Celular y Neurociencia; ArgentinaFil: Peña, Clara. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Biología Celular y Neurociencia "Prof. Eduardo de Robertis". Universidad de Buenos Aires. Facultad de Medicina. Instituto de Biología Celular y Neurociencia; Argentin

The gradient discretisation method for linear advection problems

International audienceWe adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped P1 conforming and non conforming finite element and on the hybrid finite volume method

Consistent Internal Energy Based Schemes for the Compressible Euler Equations

La deuxieme partie de ce document reprend un travail déjà exposé dans le dépot hal-01553699Numerical schemes for the solution of the Euler equations have recently been developed, which involve the discretisation of the internal energy equation, with corrective terms to ensure the correct capture of shocks, and, more generally, the consistency in the Lax-Wendroff sense. These schemes may be staggered or colocated, using either struc-tured meshes or general simplicial or tetrahedral/hexahedral meshes. The time discretization is performed by fractional-step algorithms; these may be either based on semi-implicit pressure correction techniques or segregated in such a way that only explicit steps are involved (referred to hereafter as "explicit" variants). In order to ensure the positivity of the density, the internal energy and the pressure, the discrete convection operators for the mass and internal energy balance equations are carefully designed; they use an upwind technique with respect to the material velocity only. The construction of the fluxes thus does not need any Rie-mann or approximate Riemann solver, and yields easily implementable algorithms. The stability is obtained without restriction on the time step for the pressure correction scheme and under a CFL-like condition for explicit variants: preservation of the integral of the total energy over the computational domain, and positivity of the density and the internal energy. The semi-implicit first-order upwind scheme satisfies a local discrete entropy inequality. If a MUSCL-like scheme is used in order to limit the scheme diffusion, then a weaker property holds: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L ∞ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the momentum balance. Still for the explicit scheme, with the above-mentioned MUSCL-like scheme, the same result only holds if the ratio of the time to the space step tends to zero

Finite volume schemes and Lax-Wendroff consistency

We present a (partial) historical summary of the mathematical analysis of finite differences and finite volumes methods, paying a special attention to the Lax-Richtmyer and Lax-Wendroff theorems. We then state a Lax-Wendroff consistency result for convection operators on staggered grids (often used in fluid flow simulations), which illustrates a recent generalization of the flux consistency notion designed to cope with general discrete functions

Optimal error estimates for non-conforming approximations of linear parabolic problems with minimal regularity

We consider a general linear parabolic problem with extended time boundary conditions (including initial value problems and periodic ones), and approximate it by the implicit Euler scheme in time and the Gradient Discretisation method in space; the latter is in fact a class of methods that includes conforming and nonconforming finite elements, discontinuous Galerkin methods and several others. The main result is an error estimate which holds without supplementary regularity hypothesis on the solution. This result states that the approximation error has the same order as the sum of the interpolation error and the conformity error. The proof of this result relies on an inf-sup inequality in Hilbert spaces which can be used both in the continuous and the discrete frameworks. The error estimate result is illustrated by numerical examples with low regularity of the solution