The transport of substances back and forth between surface water and groundwater is a very serious problem. We study herein the mathematical model of this setting consisting of the Stokes equations in the fluid region coupled with the Darcy equations in the porous medium, coupled across the interface by the Beavers-Joseph-Saffman conditions. We prove existence of weak solutions and give a complete analysis of a finite element scheme which allows a simulation of the coupled problem to be uncoupled into steps involving porous media and fluid flow subproblems. This is important because there are many "legacy" codes available which have been optimized for uncoupled porous media and fluid flow. Key words. coupled porous media and fluid flow, Stokes and Darcy equations, Beavers-JosephSaffman condition, weak solutions, finite element scheme, error estimates AMS subject classifications. 35Q35, 65N30, 65N15, 76D07, 76S05 PII. S00361429013927661. Introduction and the model. There are many serious problems currently facing the world in which the coupling between groundwater and surface water is important. These include questions such as predicting how pollution discharged into streams, lakes, and rivers makes its way into the water supply. This coupling is also important in technological applications involving filtration.The aim of our research is to begin the study of the following problem: an incompressible fluid in a region Ω 1 can flow both ways across an interface Γ I into a domain Ω 2 which is a porous medium saturated with the same fluid. The mathematical theory and numerical analysis of each subproblem is well developed, and reliable codes are available. Nevertheless, the mathematical theory of the coupled problem seems to be not completely understood. The model of this situation which is most accessible to large scale computations consists of the Navier-Stokes equations (or Stokes equations) in the fluid region coupled across an interface with the Darcy equations for the filtration velocity in the porous medium. This leads to mathematical difficulties arising from the coupled system of equations of different orders in different regions. See Jäger and Mikelić [16], Payne and Straughan [22] for the beginning of analytical studies of this problem. (For the Brinkman model of porous media flow this difficulty does not occur; see Jäger and Mikelić [17], Angot [1].) The second issue concerns the correct transmission conditions on the interface. The Beavers-Joseph-Saffman interface conditions [3, 25] are now well established. The third difficulty is technical: where the interface meets the other boundaries, there are incompatibilities between the imposed boundary conditions.

International audienceWe devise and analyze arbitrary-order nonconforming methods for the discretization of the viscosity-dependent Stokes equations on simplicial meshes. We keep track explicitly of the viscosity and aim at pressure-robust schemes that can deal with the practically relevant case of body forces with large curl-free part in a way that the discrete velocity error is not spoiled by large pressures. The method is inspired from the recent Hybrid High-Order (HHO) methods for linear elasticity. After elimination of the auxiliary variables by static condensation, the linear system to be solved involves only discrete face-based velocities, which are polynomials of degree $k \ge 0$, and cell-wise constant pressures. Our main result is a pressure-independent energy-error estimate on the velocity of order $(k+1)$. The main ingredient to achieve pressure-independence is the use of a divergence-preserving velocity reconstruction operator in the discretization of the body forces. We also prove an $L^2$-pressure estimate of order $(k+1)$ and an $L^2$-velocity estimate of order $(k+2)$, the latter under elliptic regularity. The local mass and momentum conservation properties of the discretization are also established. Finally, two-and three-dimensional numerical results are presented to support the analysis

We discuss numerical properties of continuous Galerkin-Petrov and discontinuous Galerkin time discretizations applied to the heat equation as a prototypical example for scalar parabolic partial differential equations. For the space discretization, we use biquadratic quadrilateral finite elements on general two-dimensional meshes. We discuss implementation aspects of the time discretization as well as efficient methods for solving the resulting block systems. Here, we compare a preconditioned BiCGStab solver as a Krylov space method with an adapted geometrical multigrid solver. Only the convergence of the multigrid method is almost independent of the mesh size and the time step leading to an efficient solution process. By means of numerical experiments we compare the different time discretizations with respect to accuracy and computational costs.

We consider the Willmore boundary value problem for surfaces of revolution where, as Dirichlet boundary conditions, any symmetric set of position and angle may be prescribed. Using direct methods of the calculus of variations, we prove existence and regularity of minimising solutions. Moreover, we estimate the optimal Willmore energy and prove a number of qualitative properties of these solutions. Besides convexity-related properties we study in particular the limit when the radii of the boundary circles converge to 0, while the "length" of the surfaces of revolution is kept fixed. This singular limit is shown to be the sphere, irrespective of the prescribed boundary angles.These analytical investigations are complemented by presenting a numerical algorithm based on C 1 -elements and numerical studies. They intensively interact with geometric constructions in finding suitable minimising sequences for the Willmore functional.

We introduce and analyze a post-processing for a family of variational space-time approximations to wave problems. The discretization in space and time is based on continuous finite element methods. The post-processing lifts the fully discrete approximations in time from continuous to continuously differentiable ones. Further, it increases the order of convergence of the discretization in time which can be be exploited nicely, for instance, for a-posteriori error control. The convergence behavior is shown by proving error estimates of optimal order in various norms. A bound of superconvergence at the discrete times nodes is included. To show the error estimates, a special approach is developed. Firstly, error estimates for the time derivative of the post-processed solution are proved. Then, in a second step these results are used to establish the desired error estimates for the post-processed solution itself. The need for this approach comes through the structure of the wave equation providing only stability estimates that preclude us from using absorption arguments for the control of certain error quantities. A further key ingredient of this work is the construction of a new time-interpolate of the exact solution that is needed in an essential way for deriving the error estimates. Finally, a conservation of energy property is shown for the post-processed solution which is a key feature for approximation schemes to wave equations. The error estimates given in this work are confirmed by numerical experiments.

We construct and analyze a discontinuous Galerkin-Petrov time discretization of a general evolution equation in a Hilbert space. The method is A-stable and exhibits an energy decreasing property. The approach consists in a continuous solution space and a discontinuous test space such that the time derivative of the discrete solution is contained in the test space. This is the key to get stability. We prove A-stability and optimal error estimates. Numerical results confirm the theoretical results.

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