A survey of mixed finite element methods

This paper is an introduction to and an overview of mixed finite element methods. It discusses the mixed formulation of certain basic problems in elasticity and hydrodynamics. It also discusses special techniques for solving the discrete problem

Stabilization arising from PGEM : a review and further developments

The aim of this paper is twofold. First, we review the recent Petrov-Galerkin enriched method (PGEM) to stabilize numerical solutions of BVP's in primal and mixed forms. Then, we extend such enrichment technique to a mixed singularly perturbed problem, namely, the generalized Stokes problem, and focus on a stabilized finite element method arising in a natural way after performing static condensation. The resulting stabilized method is shown to lead to optimal convergences, and afterward, it is numerically validated

Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids

In this paper, we consider anisotropic diffusion with decay, and the diffusivity coefficient to be a second-order symmetric and positive definite tensor. It is well-known that this particular equation is a second-order elliptic equation, and satisfies a maximum principle under certain regularity assumptions. However, the finite element implementation of the classical Galerkin formulation for both anisotropic and isotropic diffusion with decay does not respect the maximum principle. We first show that the numerical accuracy of the classical Galerkin formulation deteriorates dramatically with increase in the decay coefficient for isotropic medium and violates the discrete maximum principle. However, in the case of isotropic medium, the extent of violation decreases with mesh refinement. We then show that, in the case of anisotropic medium, the classical Galerkin formulation for anisotropic diffusion with decay violates the discrete maximum principle even at lower values of decay coefficient and does not vanish with mesh refinement. We then present a methodology for enforcing maximum principles under the classical Galerkin formulation for anisotropic diffusion with decay on general computational grids using optimization techniques. Representative numerical results (which take into account anisotropy and heterogeneity) are presented to illustrate the performance of the proposed formulation

Spreading of Kaolin and Sand Mixtures on a Horizontal Plane: Physical Experiments and SPH Numerical Modelling

open4noThe investigation of the collapse of a well-known soil volume is a simple experiment that permits to make several interesting considerations. This paper, at first, presents a brief overview of some physical experiments led to understand how the composition of a three-phase mixture influences the mass collapse. In particular, the run-out and the maximum height of the deposit are considered as two fundamental quantities for characterizing the behaviour of the mass in each test. In a second step, the experimental results obtained are used as case studies for the calibration of a mesh-less numerical model. Several simulations are carried out using the SPH-Geoflow code implementing a Bingham law to reproduce each bi-phases test. A comparison between the numerical results and the physical data permits to choose the most reliable value of the constitutive parameters for each tested case. The errors between the physical and the numerical run-out and maximum heights become the fundamental quantity to define the quality of the best simulation. Indeed, some final considerations about the relationship existing among the constitutive parameters and the kaolin content of the mixtures are reported.openBrezzi, Lorenzo; Cola, Simonetta; Gabrieli, Fabio; Gidoni, GiacomoBrezzi, Lorenzo; Cola, Simonetta; Gabrieli, Fabio; Gidoni, Giacom

Observations on degenerate saddle point problems

We investigate degenerate saddle point problems, which can be viewed as limit cases of standard mixed formulations of symmetric problems with large jumps in coefficients. We prove that they are well-posed in a standard norm despite the degeneracy. By wellposedness we mean a stable dependence of the solution on the right-hand side. A known approach of splitting the saddle point problem into separate equations for the primary unknown and for the Lagrange multiplier is used. We revisit the traditional Ladygenskaya--Babu\v{s}ka--Brezzi (LBB) or inf--sup condition as well as the standard coercivity condition, and analyze how they are affected by the degeneracy of the corresponding bilinear forms. We suggest and discuss generalized conditions that cover the degenerate case. The LBB or inf--sup condition is necessary and sufficient for wellposedness of the problem with respect to the Lagrange multiplier under some assumptions. The generalized coercivity condition is necessary and sufficient for wellposedness of the problem with respect to the primary unknown under some other assumptions. We connect the generalized coercivity condition to the positiveness of the minimum gap of relevant subspaces, and propose several equivalent expressions for the minimum gap. Our results provide a foundation for research on uniform wellposedness of mixed formulations of symmetric problems with large jumps in coefficients in a standard norm, independent of the jumps. Such problems appear, e.g., in numerical simulations of composite materials made of components with contrasting properties.Comment: 8 page

On the stability of bubble functions and a stabilized mixed finite element formulation for the Stokes problem

In this paper we investigate the relationship between stabilized and enriched finite element formulations for the Stokes problem. We also present a new stabilized mixed formulation for which the stability parameter is derived purely by the method of weighted residuals. This new formulation allows equal order interpolation for the velocity and pressure fields. Finally, we show by counterexample that a direct equivalence between subgrid-based stabilized finite element methods and Galerkin methods enriched by bubble functions cannot be constructed for quadrilateral and hexahedral elements using standard bubble functions.Comment: 25 pages, 13 figures (The previous version was compiled by mistake with the wrong style file, the current one uses amsart, and there is no difference in the text or the figures

A new data assimilation procedure to develop a debris flow run-out model

Abstract Parameter calibration is one of the most problematic phases of numerical modeling since the choice of parameters affects the model\u2019s reliability as far as the physical problems being studied are concerned. In some cases, laboratory tests or physical models evaluating model parameters cannot be completed and other strategies must be adopted; numerical models reproducing debris flow propagation are one of these. Since scale problems affect the reproduction of real debris flows in the laboratory or specific tests used to determine rheological parameters, calibration is usually carried out by comparing in a subjective way only a few parameters, such as the heights of soil deposits calculated for some sections of the debris flows or the distance traveled by the debris flows using the values detected in situ after an event has occurred. Since no automatic or objective procedure has as yet been produced, this paper presents a numerical procedure based on the application of a statistical algorithm, which makes it possible to define, without ambiguities, the best parameter set. The procedure has been applied to a study case for which digital elevation models of both before and after an important event exist, implicating that a good database for applying the method was available. Its application has uncovered insights to better understand debris flows and related phenomena

Narrating regions: New Storytelling technique helps increasing people's analysis and information sharing

Sound information at sub-national level and benchmarking of regions across national borders has increased in importance in the policy agenda of many countries due to higher integration driven by institutional processes and economic globalisation. Geovisual analytics techniques help illustrating complex data such as regional, spatiotemporal and multidimensional statistics. Interactive time-linked visual representations enable the users to simultaneously analyse relations among different variables. "OECD eXplorer", developed by NCVA in collaboration with OECD, is today a worldwide recognized web-enabled tool for visualizing and better understanding the socio-economic structure of OECD regions and their performance over time. Geovisual Analytics in the OECD explorer has so far focused more on tools to analyse regional economic performance than on methods that efficiently publish gained knowledge. Publication is indeed part of the analytical process and it could become a catalyst for discussion generating new value in a social setting. In this context, we introduce a novel storytelling that supports the editorial authoring process with the goal to advance technology critical to the sharing of information and publishing. With the introduction of this new technique, we are moving away from a clear distinction between authors and readers: The analyst can discuss with interested readers the visual discoveries which have been captured into snapshots together with descriptive text and hyperlinks. The author gets feedback from colleagues, adapts the story and publishes it using a "Vislet" that is embedded in blogs or wikis. This advanced storytelling technology applied to OECD eXplorer can therefore become a complete on-line publication to highlight recent trends and relevant disparities among OECD regions

Lowest order Virtual Element approximation of magnetostatic problems

We give here a simplified presentation of the lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic field $\mathbf{H}$ on each edge, and the vertex values of the Lagrange multiplier $p$ (used to enforce the solenoidality of the magnetic induction $\mathbf{B}=\mu\mathbf{H}$). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called "first kind N\'ed\'elec" elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions

Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM

© EDP Sciences, SMAI 2011This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in Rn (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := Rn\￣Ω. The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequality with a linear operator. Then we treat the corresponding numerical scheme and discuss an approximation of the NtD mapping with an appropriate discretization of the inverse Poincar´e-Steklov operator. In particular, assuming some abstract approximation properties and a discrete inf-sup condition, we show unique solvability of the discrete scheme and obtain the corresponding a-priori error estimate. Next, we prove that these assumptions are satisfied with Raviart- Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory