SUMMARYThe link between Mixed Finite Element (MFE) and Finite Volume (FV) methods applied to elliptic partial di erential equations has been investigated by many authors. Recently, a FV formulation of the mixed approach has been developed. This approach was restricted to 2D problems with a scalar for the parameter used to calculate uxes from the state variable gradient. This new approach is extended to 2D problems with a full parameter tensor and to 3D problems. The objective of this new formulation is to reduce the total number of unknowns while keeping the same accuracy. This is achieved by deÿning one new variable per element.For the 2D case with full parameter tensor, this new formulation exists for any kind of triangulation. It allows the reduction of the number of unknowns to the number of elements instead of the number of edges. No additional assumptions are required concerning the averaging of the parameter in heterogeneous domains. For 3D problems, we demonstrate that the new formulation cannot exist for a general 3D tetrahedral discretization, unlike in the 2D problem. However, it does exist when the tetrahedrons are regular, or deduced from rectangular parallelepipeds, and allows reduction of the number of unknowns. Numerical experiments and comparisons between both formulations in 2D show the e ciency of the new formulation.

SUMMARYDiffusion-type partial differential equation is a common mathematical model in physics. Solved by mixed finite elements, it leads to a system matrix which is not always an M-matrix. Therefore, the numerical solution may exhibit unphysical results due to oscillations. The criterion necessary to obtain an M-matrix is discussed in details for triangular, rectangular and tetrahedral elements. It is shown that the system matrix is never an M-matrix for rectangular elements and can be an M-matrix for triangular an tetrahedral elements if criteria on the element's shape and on the time step length are fulfilled. A new mass lumping scheme is developed which leads to a less restrictive criterion: the discretization must be weakly acute (all angles less than /2) and there is no constraint on the time step length. The lumped formulation of mixed hybrid finite element can be applied not only to triangular meshes but also to more general shape elements in two and three dimensions. Numerical experiments show that, compared to the standard mixed hybrid formulation, the lumping scheme avoids (or strongly reduce) oscillations and does not create additional numerical errors.

[1] In this work, we show how the use of global sensitivity analysis (GSA) in conjunction with the polynomial chaos expansion (PCE) methodology can provide relevant information for the interpretation of transport experiments in laboratory-scale heterogeneous porous media. We perform GSA by calculating the Sobol indices, which provide a variance-based importance measure of the effects of uncertain parameters on the output of a chosen interpretive transport model. The choice of PCE has the following two benefits: (1) it provides the global sensitivity indices in a straightforward manner, and (2) PCE can serve as a surrogate model for the calibration of parameters. The coefficients of the PCE are computed by probabilistic collocation. The methodology is applied to two nonreactive transport experiments available in the literature, while considering both transient and pseudo steady state transport regimes. This method allows a rigorous investigation of the relative effects and importance of different uncertain quantities, which include boundary conditions as well as porous medium hydraulic and dispersive parameters. The parameters that are most relevant to depicting the system's behavior can then be evaluated. In addition, one can assess the space-time distribution of measurement points, which is the most influential factor for the identifiability of parameters. Our work indicates that these methods can be valuable tools in the proper design of model-based transport experiments.

Polynomial chaos expansions are frequently used by engineers and modellers for uncertainty and sensitivity analyses of computer models. They allow representing the input/output relations of computer models. Usually only a few terms are really relevant in such a representation. It is a challenge to infer the best sparse polynomial chaos expansion of a given model input/output data set. In the present article, sparse polynomial chaos expansions are investigated for global sensitivity analysis of computer model responses. A new Bayesian approach is proposed to perform this task, based on the Kashyap information criterion for model selection. The efficiency of the proposed algorithm is assessed on several benchmarks before applying the algorithm to identify the most relevant inputs of a double-diffusive convection model.