Finite element pressure stabilizations for incompressible flow problems

Discretizations of incompressible flow problems with pairs of finite element spaces that do not satisfy a discrete inf-sup condition require a so-called pressure stabilization. This paper gives an overview and systematic assessment of stabilized methods, including the respective error analysis

Penalty-free Nitsche method for interface problems

Nitsche’s method is a penalty-based method to weakly enforce boundary conditions in the finite element method. In this paper, we present a penalty free version of Nitsche’s method to impose interface coupling in the framework of unfitted domain decomposition. Unfitted domain decomposition is understood in the sense that the interface between the domains can cross elements of the mesh arbitrarily. The pure diffusion problem with discontinuous material parameters is considered for the theoretical study, we show the convergence of the L2 and H1-error for high contrast in the diffusivities. Then, we give the corresponding numerical results for the pure diffusion problem, additionally we consider the Stokes problem. We compare the performance of the penalty free method with the more classical symmetric and nonsymmetric Nitsche’s methods for different cases, including for the error generated in the interface fluxes

Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods

International audienceWe devise a space-time tensor method for the low-rank approximation of linear parabolic evolution equations. The proposed method is a stable Galerkin method, uniformly in the discretization parameters, based on a Minimal Residual formulation of the evolution problem in Hilbert--Bochner spaces. The discrete solution is sought in a linear trial space composed of tensors of discrete functions in space and in time and is characterized as the unique minimizer of a discrete functional where the dual norm of the residual is evaluated in a space semi-discrete test space. The resulting global space-time linear system is solved iteratively by a greedy algorithm. Numerical results are presented to illustrate the performance of the proposed method on test cases including non-selfadjoint and time-dependent differential operators in space. The results are also compared to those obtained using a fully discrete Petrov--Galerkin setting to evaluate the dual residual norm

Epidemiological Forecasting with Model Reduction of Compartmental Models. Application to the COVID-19 Pandemic

International audienceWe propose a forecasting method for predicting epidemiological health series on a two-week horizon at regional and interregional resolution. The approach is based on the model orderreduction of parametric compartmental models and is designed to accommodate small amounts ofsanitary data. The efficiency of the method is shown in the case of the prediction of the number ofinfected people and people removed from the collected data, either due to death or recovery, duringthe two pandemic waves of COVID-19 in France, which took place approximately between Februaryand November 2020. Numerical results illustrate the promising potential of the approac

Fictitious domain method with boundary value correction using penalty-free Nitsche method

In this paper, we consider a fictitious domain approach based on a Nitsche type method without penalty. To allow for high order approximation using piecewise affine approximation of the geometry we use a boundary value correction technique based on Taylor expansion from the approximate to the physical boundary. To ensure stability of the method a ghost penalty stabilization is considered in the boundary zone. We prove optimal error estimates in the H1-norm and estimates suboptimal b