Proof of the Double Bubble Conjecture in R^n

The least-area hypersurface enclosing and separating two given volumes in R^n is the standard double bubble.Comment: 20 pages, 22 figure

A Covolume Method Based on Rotated Bilinears for the Generalized Stokes Problem

We introduce a covolume or marker and cell (MAC) method for approximating the generalized Stokes problem on an axiparallel domain. Two grids are needed, the primal grid made up of rectangles and the dual grid of quadrilaterals. The velocity is approximated by nonconforming rotated bilinear elements with degrees of freedom at midpoints of rectangular elements and the pressure by piecewise constants. The error in the velocity in the Hh norm and the pressure in the L2 norm are of first order, provided that the exact velocity is in H2 and the exact pressure in H1

Mixed Covolume Methods on Rectangular Grids for Elliptic Problems

We consider a covolume method for a system of first order PDEs resulting from the mixed formulation of the variable-coefficient-matrix Poisson equation with the Neumann boundary condition. The system may be used to represent the Darcy law and the mass conservation law in anisotropic porous media flow. The velocity and pressure are approximated by the lowest order Raviart-Thomas space on rectangles. The method was introduced by Russell [Rigorous Block- centered Discretizations on Irregular Grids: Improved Simulation of Complex Reservoir Systems, Reservoir Simulation Research Corporation, Denver, CO, 1995] as a control-volume mixed method and has been extensively tested by Jones [A Mixed Finite Volume Elementary Method for Accurate Computation of Fluid Velocities in Porous Media, University of Colorado at Denver, 1995] and Cai et al. [Computational Geosciences, 1 (1997), pp. 289-345]. We reformulate it as a covolume method and prove its first order optimal rate of convergence for the approximate velocities as well as for the approximate pressures

Flux Recovery from Primal Hybrid Finite Element Methods

A flux recovery technique is introduced and analyzed for the computed solution of the primal hybrid finite element method for second-order elliptic problems. The recovery is carried out over a single element at a time while ensuring the continuity of the flux across the interelement edges and the validity of the discrete conservation law at the element level. Our construction is general enough to cover all degreesof polynomialsand gridsof triangular or quadrilateral type. We illustrate the principle using the Raviart–Thomas spaces, but other well-known related function spaces such as the Brezzi–Douglas–Marini (BDM) or Brezzi–Douglas–Fortin–Marini (BDFM) space can be used as well. An extension of the technique to the nonlinear case is given. Numerical results are presented to confirm the theoretical results

Mixed Covolume Methods for Elliptic Problems on Triangular Grids

We consider a covolume or finite volume method for a system of first-order PDEs resulting from the mixed formulation of the variable coefficient-matrix Poisson equation with the Neumann boundary condition. The system may represent either the Darcy law and the mass conservation law in anisotropic porous media flow, or Fourier law and energy conservation. The velocity and pressure are approximated by the lowest order Raviart-Thomas space on triangles. We prove its first-order optimal rate of convergence for the approximate velocities in the L2-and H(div; Q)-norms as well as for the approximate pressures in the L2-norm. Numerical experiments are included