A review of recent advances in discretization methods, a posteriori error analysis, and adaptive algorithms for numerical modeling in geosciences

International audienceTwo research subjects in geosciences which lately underwent significant progress are treated in this review. In the first part we focus on one key ingredient for the numerical approximation of the Darcy flow problem, namely the discretization of diffusion terms on general polygonal/polyhedral meshes. We present different schemes and discuss in detail their fundamental numerical properties such as stability, consistency, and robustness. The second part of the paper is devoted to error control and adaptivity for model geosciences problems. We present the available a posteriori estimates guaranteeing the maximal overall error and show how the different error components can be identified. These estimates are used to formulate adaptive stopping criteria for linear and nonlinear solvers, time step choice adjustment, and adaptive mesh refinement. Numerical experiments illustrate such entirely adaptive algorithms

The DPG-star method

This article introduces the DPG-star (from now on, denoted DPG$^*$) finite element method. It is a method that is in some sense dual to the discontinuous Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG$^*$ methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddle-point problems. The analyses of the two problems have many common elements. Comparison to other methods in the literature round out the newly garnered perspective. Notably, DPG$^*$ and DPG methods can be seen as generalizations of $\mathcal{L}\mathcal{L}^\ast$ and least-squares methods, respectively. A priori error analysis and a posteriori error control for the DPG$^*$ method are considered in detail. Reports of several numerical experiments are provided which demonstrate the essential features of the new method. A notable difference between the results from the DPG$^*$ and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable

Vertex-based Compatible Discrete Operator schemes on polyhedral meshes for advection-diffusion equations

International audienceWe 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

Finite element pressure stabilizations for incompressible flow problems

Discretizations of incompressible flow problems with pairs of finite element spaces that do not satisfy a discrete inf-sup condition require a so-called pressure stabilization. This paper gives an overview and systematic assessment of stabilized methods, including the respective error analysis

Bridging the Hybrid High-Order and Virtual Element methods

International audienceWe present a unifying viewpoint at Hybrid High-Order and Virtual Element methods on general polytopal meshes in dimension $2$ or $3$, both in terms of formulation and analysis. We focus on a model Poisson problem. To build our bridge, (i) we transcribe the (conforming) Virtual Element method into the Hybrid High-Order framework, and (ii) we prove $H^m$ approximation properties for the local polynomial projector in terms of which the local Virtual Element discrete bilinear form is defined. This allows us to perform a unified analysis of Virtual Element/Hybrid High-Order methods, that differs from standard Virtual Element analyses by the fact that the approximation properties of the underlying virtual space are not explicitly used. As a complement to our unified analysis, we also study interpolation in local virtual spaces, shedding light on the differences between the conforming and nonconforming cases

Stabilized continuous and discontinuous Galerkin techniques for Darcy flow

We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain, leading to stabilized methods with different stability and convergence properties. These stabilized methods mimic different possible functional settings of the continuous problem. The optimal method depends on the velocity and pressure approximation order. They also involve a subgrid projector that can be either the identity (when applied to finite element residuals) or can have an image orthogonal to the finite element space. In particular, we have designed a new stabilized method that allows the use of piecewise constant pressures. We consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations. All these methods have been analyzed, proving stability and convergence results. In some cases, duality arguments have been used to obtain error bounds in the L2-norm

Local Finite Element Approximation of Sobolev Differential Forms

We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Cl\'ement interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert Lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.Comment: 22 pages. Comments welcom