A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes

In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady non-linear Leray-Lions problems. The proposed method has several assets, including the support for arbitrary approximation orders and general polytopal meshes. This is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case. The convergence analysis is carried out using a compactness technique. Extending this technique to HHO methods has prompted us to develop a set of discrete functional analysis tools whose interest goes beyond the specific problem and method addressed in this work: (direct and) reverse Lebesgue and Sobolev embeddings for local polynomial spaces, $L^{p}$-stability and $W^{s,p}$-approximation properties for $L^{2}$-projectors on such spaces, and Sobolev embeddings for hybrid polynomial spaces. Numerical tests are presented to validate the theoretical results for the original method and variants thereof

Reflected BSDEs when the obstacle is not right-continuous and optimal stopping

In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is established by using some tools from the general theory of processes such as Mertens decomposition of optional strong (but not necessarily right-continuous) supermartingales, some tools from optimal stopping theory, as well as an appropriate generalization of It{\^o}'s formula due to Gal'chouk and Lenglart. In the second part of the paper, we provide some links between the RBSDE studied in the first part and an optimal stopping problem in which the risk of a financial position $\xi$ is assessed by an $f$-conditional expectation $\mathcal{E}^f(\cdot)$ (where $f$ is a Lipschitz driver). We characterize the "value function" of the problem in terms of the solution to our RBSDE. Under an additional assumption of left upper-semicontinuity on $\xi$, we show the existence of an optimal stopping time. We also provide a generalization of Mertens decomposition to the case of strong $\mathcal{E}^f$-supermartingales

On Lipschitz properties of generated aggregation functions

This article discusses Lipschitz properties of generated aggregation functions. Such generated functions include triangular norms and conorms, quasi-arithmetic means, uninorms, nullnorms and continuous generated functions with a neutral element. The Lipschitz property guarantees stability of aggregation operations with respect to input inaccuracies, and is important for applications. We provide verifiable sufficient conditions to determine when a generated aggregation function holds the k-Lipschitz property, and calculate the Lipschitz constants of power means. We also establish sufficient conditions which guarantee that a generated aggregation function is not Lipschitz. We found the only 1-Lipschitz generated function with a neutral element e &isin;]0, 1[.<br /

Compatible finite element spaces for geophysical fluid dynamics

Compatible finite elements provide a framework for preserving important structures in equations of geophysical uid dynamics, and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical uid dynamics, including the application to the nonlinear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarise analytic results about dispersion relations and conservation properties, and present new results on approximation properties in three dimensions on the sphere, and on hydrostatic balance properties