Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the $L^2$-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier--Stokes equations. The novel error estimators only take the $\mathrm{curl}$ of the right-hand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure-dominant situations. This is also confirmed by some numerical examples with the novel pressure-robust modifications of the Taylor--Hood and mini finite element methods

A Low Mach Number Model for Moist Atmospheric Flows

We introduce a low Mach number model for moist atmospheric flows that accurately incorporates reversible moist processes in flows whose features of interest occur on advective rather than acoustic time scales. Total water is used as a prognostic variable, so that water vapor and liquid water are diagnostically recovered as needed from an exact Clausius--Clapeyron formula for moist thermodynamics. Low Mach number models can be computationally more efficient than a fully compressible model, but the low Mach number formulation introduces additional mathematical and computational complexity because of the divergence constraint imposed on the velocity field. Here, latent heat release is accounted for in the source term of the constraint by estimating the rate of phase change based on the time variation of saturated water vapor subject to the thermodynamic equilibrium constraint. We numerically assess the validity of the low Mach number approximation for moist atmospheric flows by contrasting the low Mach number solution to reference solutions computed with a fully compressible formulation for a variety of test problems

β-NMF AND SPARSITY PROMOTING REGULARIZATIONS FOR COMPLEX MIXTURE UNMIXING. APPLICATION TO 2D HSQC NMR

International audienceIn Nuclear Magnetic Resonance (NMR) spectroscopy, an efficient analysis and a relevant extraction of different molecule properties from a given chemical mixture are important tasks, especially when processing bidimensional NMR data. To that end, using a blind source separation approach based on a vari-ational formulation seems to be a good strategy. However, the poor resolution of NMR spectra and their large dimension require a new and modern blind source separation method. In this work, we propose a new variational formulation for blind source separation (BSS) based on a β-divergence data fidelity term combined with sparsity promoting regularization functions. An application to 2D HSQC NMR experiments illustrates the interest and the effectiveness of the proposed method whether in simulated or real cases

An algebraic subgrid scale finite element method for the convected Helmholtz equation in two dimensions with applications in aeroacoustics

An algebraic subgrid scale finite element method formally equivalent to the Galerkin Least-Squares method is presented to improve the accuracy of the Galerkin finite element solution to the two-dimensional convected Helmholtz equation. A stabilizing term has been added to the discrete weak formulation containing a stabilization parameter whose value turns to be the key for the good performance of the method. An appropriate value for this parameter has been obtained by means of a dispersion analysis. As an application, we have considered the case of aerodynamic sound radiated by incompressible flow past a two-dimensional cylinder. Following Lighthill’s acoustic analogy, we have used the time Fourier transform of the double divergence of the Reynolds stress tensor as a source term for the Helmholtz and convected Helmholtz equations and showed the benefits of using the subgrid scale stabilization

Smoothed Particle Magnetohydrodynamics III. Multidimensional tests and the div B = 0 constraint

In two previous papers (Price & Monaghan 2004a,b) (papers I,II) we have described an algorithm for solving the equations of Magnetohydrodynamics (MHD) using the Smoothed Particle Hydrodynamics (SPH) method. The algorithm uses dissipative terms in order to capture shocks and has been tested on a wide range of one dimensional problems in both adiabatic and isothermal MHD. In this paper we investigate multidimensional aspects of the algorithm, refining many of the aspects considered in papers I and II and paying particular attention to the code's ability to maintain the div B = 0 constraint associated with the magnetic field. In particular we implement a hyperbolic divergence cleaning method recently proposed by Dedner et al. (2002) in combination with the consistent formulation of the MHD equations in the presence of non-zero magnetic divergence derived in papers I and II. Various projection methods for maintaining the divergence-free condition are also examined. Finally the algorithm is tested against a wide range of multidimensional problems used to test recent grid-based MHD codes. A particular finding of these tests is that in SPMHD the magnitude of the divergence error is dependent on the number of neighbours used to calculate a particle's properties and only weakly dependent on the total number of particles. Whilst many improvements could still be made to the algorithm, our results suggest that the method is ripe for application to problems of current theoretical interest, such as that of star formation.Comment: Here is the latest offering in my quest for a decent SPMHD algorithm. 26 pages, 15 figures, accepted for publication in MNRAS. Version with high res figures available from http://www.astro.ex.ac.uk/people/dprice/pubs/spmhd/spmhdpaper3.pd

Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure-independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the L2-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the NavierStokes equations. The novel error estimators only take the curl of the righthand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure-dominant situations. This is also confirmed by some numerical examples with the novel pressure-robust modifications of the TaylorHood and mini finite element methods