The theoretical analysis of mesh-adaptive methods is a very active field of research. We have generalized our previous results concerning optimality of adaptive methods to nonconforming finite elements . Our results include the error due to iterative solution of the system matrices by means of a simple stopping criterion related to the error estimator. The main difficulty was the treatment of the nonconformity which leads to a perturbation of the orthogonality relation at the heart of the proofs for conforming finite elements. We have been able to extend this result to the Stokes equations, considering different lowest-order nonconforming finite elements on triangular and quadrilateral meshes .

In we have shown that the smallness assumption required in all former proofs of optimality of adaptive finite element methods can be overcome, at least in some situations.

Finally, we have shown optimality of a new goal-oriented method in .

Our theoretical studies, which are motivated by the aim to develop better adaptive algorithms, have been accompanied by software implementation with the Concha library, see Section  . It hopefully opens the door to further theoretical and experimental studies.

The original formulation of NXFEM is based on the doubling of elements. In some situations, as the case of a moving interface, it is computationally more convenient to have a method with local enrichment, as for the standard XFEM. In we have developed such an approach based on NXFEM. We have developed an hierarchical formulation for a fictitious domain formulation in

One of the technical difficulties is the simultaneous robustness of the method with respect to the size of the intersection of a mesh cell with the interface and with respect to the discontinuous diffusion parameters. This is the subject of the thesis of Nelly Barrau, supervised by Robert Luce and Eric Dubach (LMAP).

We have developed a new discontinuous Galerkin scheme for the Stokes equations and corresponding three-fields formulation. In this work, which is part of the Phd Thesis of Julie Joie, we introduce a modification of the stabilization term in the standard DG-IP method. This allows for a cheaper implementation and has a more robust behavior with respect to the stabilization parameter; we have shown convergence towards the solution of non-conforming finite element methods for linear, quadratic and cubic polynomial degrees. This scheme has been extended to the three-field formulation of the Stokes problem, which is a further step towards the polymer project of Section  . Since it is well known that the non-conforming finite element approximations do not verify the discrete Korn inequality, an appropriate further stabilization term is introduced, see  .

We have developed a new stabilized finite element formulation based on implicit penalization of the singularities of higher-order derivatives of continuous finite element spaces in . It has been applied to the convection-diffusion problem as well as to the incompressible Euler equations.

Mesh adaptivity is nowadays an essential tool in numerical simulations; in order to achieve it, reliable and efficient, easily computable a posteriorierror estimators are needed. Such estimators obtained by reconstructing locally conservative fluxes in the Raviart-Thomas finite element space have been largely employed in the past years.

We have so far considered the convection-diffusion equation and proposed a unified framework for several finite element approximations (conforming, nonconforming and discontinuous Galerkin). The main advantage of our approach is to use, contrarily to the existing references, only the primal mesh for the flux reconstruction, which presents certain facilities from a computational point of view.

For this purpose, the construction of the $H\left(\mathrm{div}\right)$-vector involved in the error estimator is inspired by the hypercircle method cf.  and is achieved on patches, which may overlap. A patch depends on the type of the employed finite elements and is defined as the support of a basis function.

Our first results were presented in  . We are working on the extension to higher-order approximations, to quadrilateral meshes and to other model problems.

Most practical applications involve parameters $q={\left(q_i\right)}_{1\le i\le N}$of different origins: physical (viscosity, heat conduction), modeling (computational domain, boundary conditions) and numerical (mesh, stabilization parameters, stopping criteria, values of a turbulence model). Numerical simulations can provide information related to the (first order) sensitivity of a quantity of physical interest $I\left(q\right)$with respect to different parameters: $\partial I/\partial q_i$. Their computation can help to validate the physical model, to explain unexpected behaviour and also to guide efforts to improve both the physical and the computational models.

A posteriorierror estimates for the functional itself, for fixed values of the parameters $q$, are well-known, cf. for example  where a goal-oriented error control is achieved by introducing an adjoint problem. Our goal is to provide a general framework for the a posteriorierror estimation of sensitivities $\partial I/\partial q_i-\partial I_h/\partial q_i$, which has not been given yet in the literature.

So far, we have applied the proposed method to the computation of the Nusselt number measuring the efficiency of a cooling process, described in the project Optimal. A cold liquid is injected in a annular domain through several inlets in order to cool a heated interior stator.

First numerical results, including adaptation with respect to the functional and to the sensitivity, have been carried out with the library Concha. They have been presented in  , . In Figure  one may see the computed temperature and velocity field, while the a posteriorierror estimator for the sensitivity of the Nusselt number with respect to the inflow speed at the right-hand side inlet, as well as the adapted mesh, are given in Figure  .

In the future, several important aspects related to the adaptive method are still to be investigated such as design of an appropriate adaptive algorithm, proof of its convergence and optimality etc.

We have continued our activities with respect to numerical simulations of polymer flows.

We have further validated  the code on both triangular and quadrilateral 2D meshes, by using a mixed non-conforming/DG method. In the case of Giesekus model, comparisons of velocity profile with semi-analytical solutions for Poiseuille flow  and with experimental data  for the 4:1 contraction are respectively shown in Figures and ).

The quadrilateral scheme needs an additional stabilization term, in order to ensure uniform consistency and stability for the underlying linear problem. It has been tested in particular on the benchmark case of flow around a 2D cylinder, for which our code converged for high values of the Weissenberg number (We > 70). We have computed the drag and compared it, see Figure  , with numerical data found in the literature , , , in the case of the Oldroyd-B model.

The system associated to this quasi-linear model has been solved on rather fine meshes (of about 10 ${}^6$elements) thanks to a multigrid solver based on a Vanka type preconditionner. We have worked on the extension of the multigrid method to more complex and more realistic polymer models, involving nonlinear terms in the constitutive law such that Phan Thien-Tanner and Giesekus models.

The unsteady case has also been treated for different geometries.

The parallelization of the library is ongoing work as described in the section  ; first tests have been carried on in the case of viscoelastic liquids.

In order to better describe real polymer flows, the thermal aspects have to be taken into account. As a first step, the coupling of the flow with the energy equation in the Newtonian case was considered in  , by using a conservative variable. Our next objective is to simulate anisothermal viscoelastic flows, including typical effects such that thermo-dependent viscosity and viscous heating.

With these tools, the long-term goal is to successively build up robust and efficient software in order to tackle design problems.

Moreover, we intend to consider other models of non-Newtonian fluids employed in other application domains such as biomedecine.

The stability and robustness with respect to physical parameters of numerical schemes for polymer flows is a challenging question. Indeed, most algorithms encounter serious convergence problems for large Weissenberg numbers. In the recent years, this issue has been asscociated to the discrete positive definiteness of the so-called conformation tensor. It seems therefore essential for the numerical simulations to employ positivity preserving schemes.

In order to develop such schemes, we have adopted the approach proposed by Lee and Xu  in the case of the quasi-linear Oldroyd-B model, based on the similarities between its constitutive equation and differential Riccati equations. We have applied it to a more general matrix equation; a typical example is the nonlinear Giesekus constitutive law. In agreement with our code (see Section  ), we have discretized it by a discontinuous Galerkin method combined with an upwinding of the transport terms, whereas the approach of Lee and Xu relies essentially on the characteristics method.

We have shown that a modification of Newton's method yields a monotone and positive scheme, under certain hypothesis, and we have applied this study to both Oldroyd-B and Giesekus polymer models. This allowed us to better understand and explain the better behaviour of the latter for large Weissenberg numbers. These results have been presented in , .

We are working on several challenging questions such as the extension to other discretization methods, the improvement of the iterative method or the derivation of energy estimates for the coupled system.

Over the past years, significant advances have been made in developing discontinuous Galerkin finite element methods (DGFEM) for applications in fluid flow and heat transfer. Certain features of the method have made it attractive as an alternative to other popular methods such as finite volume and more convenient finite element methods in thermal fluid engineering analyses. The DGFEM has been used successfully to solve hyperbolic systems of conservation laws. It makes use of the same local function space as the continuous method, but with relaxed continuity at inter-element boundaries. Since it uses discontinuous piecewise polynomial bases, the discretization is locally conservative and in the considered lowest-order case, the method preserves the maximum principle for scalar equations.

One of the challenges in Computational Fluid Dynamic (CFD) is to obtain as accurate as possible the solution of the problem under consideration at very low cost in terms of computational time. So our principal work is to find some relevant and robust strategies and technics of meshes adaptation in order to concentrate just the calculation where there are physical phenomena to capture. From Industrial point of view, the aim is to get the stationary solution as quick as possible with as much accuracy as possible. The main limitation of these results in CFD concern the underlying models: for example, nearly nothing seems to be known for (even linear) first-order systems or for realistic nonlinear equations. We therefore have developed different modern techniques, especially adaptive methods, to tackle this kind of problems in compressible CFD. The strategy is to iteratively improve the quality of the approximate solutions based on computed information (a posteriori error analysis). In this way, a sequence of locally refined meshes is constructed, which allows for better efficiency as compared to more classical approaches in the presence of different kind of singularities. The main goal is to improve the aerodynamical design process for complex configurations by significantly reducing the time from geometry to solution at engineering-required accuracy using high-order adaptive methods.

One of our strategies of refinement is based on the creation of hanging nodes commonly called non-conforming refinement. The figures  show superposition of two kinds of meshes. One is a non-conforming refined mesh (black color) and the other one is the initial grid (red color) on which the refinement has been performed. It shows the technic of cutting the cells where singularities occur in the scramjet inlet.

The mesh adaptation is designed using some criteria as a posteriori error estimates. We have designed criteria based on the calculation of the jump of physical quantities like density, pressure, entropy, temperature and mach number at the inter-element. This criteria seems to be a very good indicator for the mesh adaptation. Figure  is the comparison of isoline of the density in scramjet internal flow at mach 3 of the initial mesh, the third and the sixth mesh after refinement.The indicator used is the density jump. It shows the impact and the accuracy of the solution obtained after the sixth iteration of the refinement.

The figure  shows the streamlines of the density in the scramjet inlet after the seventh iteration. This shows how the adaptation depicts almost clearly and accurately the shock waves and the expansion waves and their interactions in the domain.

Figure  represent the density isolines of a flow past cylinder test case using the non-conforming mesh adaptation with quadrangular an triangular girds.

We have also settled another indication which is hierarchical. It measures the difference of $g_h$with the physical quantity $g_{h/2}$obtained by computation on a globally refined mesh $h/2$. This allows us the make comparison with the previous indicator. The case test considered for this comparison is an external flows past a cylinder airfoil at fixed free stream conditions : $M_{\infty }=3$. The result is quite surprising the way one type of indicator can capture phenomenon that are not capture by the another one. In fact the hierarchical indicator seems to capture recirculation downstream to the obstacle which was not capture by the jump indicator (see figure   )

We compare the computational time between a non-conforming mesh refinement and a globally mesh refined with nearly the same amount of cells. The meshes contain quadrangles or triangles. We can observe trough the following tables that the adapted meshes wether triangular or quadrangular meshes allow to save 20 to 90 times the computational time than the normal globally refined mesh. (see tables 1 and 2)

In table 1, the gain in time is 35 times in quadrangular grid case and 90 times triangular ones and in table 2, the gain in time: 18 times in quadrangular grid case and 58 times triangular ones. So one can say that the adaptive mesh with the strategies and technics we have settled are efficient and robust in capturing physical phenomenon at a very reasonable low cost.

In concluding, the procedure of refinement permit to save computational time and have good accuracy of the approximated solution computed. Our focus is to continue the improve our methods and strategies in order to meet the requirement of accuracy, robustness and efficiency. Many other works are in hand such as slope limiters for high-order Discontinous Galerkin, low mach number computation with some remarkable approaches.