Partial differential equationsWell-posedness of the scalar and the vector advection-reaction problems in Banach graph spaces Analyse des problèmes d'advection-réaction scalaire et vectoriel dans les espaces de Banach du graphe

We devise and analyze an edge-based scheme on polyhedral meshes to approximate a vector advection-reaction problem. The well-posedness of the discrete problem is analyzed first under the classical positivity hypothesis of Friedrichs’ systems that requires a lower bound on the lowest eigenvalue of some tensor depending on the model parameters. We also prove stability when the lowest eigenvalue is null or even slightly negative if the mesh size is small enough. A priori error estimates are established for solutions in W1,q(Ω) with q ∈ ((3/2),2]. Numerical results are presented on three-dimensional polyhedral meshes.

Abstract:We devise and analyze vertex-based, Péclet-robust, lowest-order schemes for advection-diffusion equations that support polyhedral meshes. The schemes are formulated using Compatible Discrete Operators (CDO), namely, primal and dual discrete differential operators, a discrete contraction operator for advection, and a discrete Hodge operator for diffusion. Moreover, discrete boundary operators are devised to weakly enforce Dirichlet boundary conditions. The analysis sheds new light on the theory of Friedrichs' operators at the purely algebraic level. Moreover, an extension of the stability analysis hinging on inf-sup conditions is presented to incorporate divergence-free velocity fields under some assumptions. Error bounds and convergence rates for smooth solutions are derived and numerical results are presented on three-dimensional polyhedral meshes.

The present paper is concerned with a class of penalized Signorini problems also called normal compliance models. These nonlinear models approximate the Signorini problem and are characterized both by a penalty parameter ε and by a "power parameter" α ≥ 1, where α = 1 corresponds to the standard penalization. We choose a continuous conforming linear finite element approximation in space dimensions d = 2, 3 to obtain an L 2 -error estimate of order h 2 when d = 2, α = 2, ε ≥ θh (θ large enough) and when the solution is W 2,3 -regular. A similar estimate is obtained when d = 3 under slightly more restrictive assumptions on ε.

We devise and analyze vertex-based schemes on polyhedral meshes to approximate advection-reaction equations. Error estimates of order O(h 3/2) are established in the discrete inf-sup stability norm which includes the mesh-dependent weighted advective derivative. The two key ingredients are a local polyhedral reconstruction map leaving affine polynomials invariant, and a local design of stabilization whereby gradient jumps are only penalized across some subfaces in the interior of each mesh cell. Numerical results are presented on three-dimensional polyhedral meshes.