local discontinuous galerkin methods for the stokes system in this paper we introduce and analyze local discontinuous galerkin methods for the stokes system for a class of shape regular meshes with hanging nodes we derive a priori estimates for the l2 norm of the errors in the velocities and the pressure we show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k 1 for the pressure for any k ge1 we also consider the case in which all the unknowns are approximated with polynomials of degree k and show that although the orders of convergence remain the same the method is more efficient numerical experiments verifying these facts are displayed

discontinuous finite volume methods for the stationary stokes darcy problem summary in this paper we analyze discontinuous finite volume methods for the stationary stokes darcy problem that models coupled fluid flow and porous media flow the discontinuous finite volume methods are combinations of finite volume method and discontinuous galerkin method with three interior penalty types incomplete symmetric nonsymmetric and symmetric briefly using discontinuous functions as trial functions in the finite volume method optimal error estimates in broken h1 norm are obtained for the three discontinuous finite volume methods optimal error estimates in the standard l2 norm are derived for the symmetric interior penalty discontinuous finite volume method numerical experiments are presented to confirm the theoretical results with non matching meshes across the common interface of stokes region and darcy region copyright 2015 john wiley sons ltd

a crank nicolson discontinuous finite volume element method for a coupled non stationary stokes darcy problem abstract in this paper we propose a fully discrete scheme which is based on discontinuous finite volume element methods in space and a crank nicolson discretization in time and we obtain the solution at the first time step by applying a first order backward euler method to solve the coupling of time dependent stokes darcy problem the proposed numerical method is established on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head whereas the pressure is approximated by the piecewise constant elements then the unique solvability of the approximate solution for the discrete problem is derived in addition the optimal error estimates of the full discretization in standard l 2 norm and broken h 1 norm are obtained finally a series of numerical experiments are provided to illustrate the features of the proposed method such as the optimal accuracy orders adaptive mesh refinement mass conservation capability to conveniently deal with complicated geometries with different types of boundary conditions and the applicability to the problems with realistic parameters in particular the last numerical experiment shows the capability of the method to deal with complicated networks of fractures in porous media domain it encompasses both fractures with high permeability which act as the prior flow conduit and fractures with low permeability as a flow barrier

analysis and convergence of a covolume method for the generalized stokes problem we introduce a covolume or mac like method for approximating the generalized stokes problem two grids are needed in the discretization a triangular one for the contimlity equation and a quadrilateral one for the momentum equation the velocity is approximated using nonconforming piecewise linears and the pressure piecewise constants error in the l 2 norm for the pressure and error in a mesh dependent h 1 norm as well as in the l 2 norm for the velocity are shown to be of first order provided that the exact velocity is in h 2 and the true pressure in h i we also introduce the concept of a network model into the discretized linear system so that an efficient pressure recovering technique can be used to simplify a great deal the computational work involved in the augmented lagrangian method given is a very general decomposition condition under which this technique is applicable to other fluid problems that can be formulated as a saddle point problem

analysis of a finite volume element method for the stokes problem in this paper we propose a stabilized conforming finite volume element method for the stokes equations on stating the convergence of the method optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations a superconvergence result is also derived by using a postprocessing projection method in particular the stabilization of the continuous lowest equal order pair finite volume element discretization is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries finally some numerical experiments that confirm the predicted behavior of the method are provided

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2

discontinuous approximation of viscous two phase flow in heterogeneous porous media runge kutta discontinuous galerkin rkdg and discontinuous finite volume element dfve methods are applied to a coupled flow transport problem describing the immiscible displacement of a viscous incompressible fluid in a non homogeneous porous medium the model problem consists of nonlinear pressure velocity equations assuming brinkman flow coupled to a nonlinear hyperbolic equation governing the mass balance saturation equation the mass conservation properties inherent to finite volume based methods motivate a dfve scheme for the approximation of the brinkman flow in combination with a rkdg method for the spatio temporal discretization of the saturation equation the stability of the uncoupled schemes for the flow and for the saturation equations is analyzed and several numerical experiments illustrate the robustness of the numerical method

a new backtracking inexact bfgs method for symmetric nonlinear equations a bfgs method in association with a new backtracking line search technique is presented for solving symmetric nonlinear equations the global and superlinear convergences of the given method are established under mild conditions preliminary numerical results show that the proposed method is better than the normal technique for the given problems

a discontinuous finite volume method for a coupled fracture model abstract in this paper we present a discontinuous finite volume method for a hybrid dimensional coupled fracture model on triangular meshes in matrix and fracture the pressure is approximated by the discontinuous and continuous piecewise linear elements and in order to retain local conservation property the velocity is sought in the discontinuous lowest order raviart thomas element space and piecewise linear element space the proposed numerical scheme is only associated with the pressure so we avoid solving an indefinite saddle point problem in addition the discontinuous method eliminates the continuity of approximate functions across the interelement boundary so it has the advantage of high localizability and easy handling of complicated geometries optimal order error estimates for the pressure and velocity are proved and numerical experiments are carried out to confirm our theoretical analysis

a discontinuous mixed covolume method for elliptic problems we develop a discontinuous mixed covolume method for elliptic problems on triangular meshes an optimal error estimate for the approximation of velocity is obtained in a mesh dependent norm first order l 2 error estimates are derived for the approximations of both velocity and pressure

convergence of the discontinuous finite volume method for elliptic problems with minimal regularity this paper investigates convergence of the discontinuous finite volume method dfvm under minimal regularity assumptions on solutions of second order elliptic boundary value problems conventional analysis requires the solutions to be in sobolev spaces h 1 s s 12 here we assume the solutions are in h 1 s s 0 and employ the techniques developed in gudi 2010 18 20 to derive error estimates in a mesh dependent energy norm and the l 2 norm for dfvm the theoretical estimates are illustrated by numerical results which include problems with corner singularity and intersecting interfaces

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

discontinuous finite volume element discretization for coupled flow transport problems arising in models of sedimentation the sedimentation consolidation and flow processes of a mixture of small particles dispersed in a viscous fluid at low reynolds numbers can be described by a nonlinear transport equation for the solids concentration coupled with the stokes problem written in terms of the mixture flow velocity and the pressure field here both the viscosity and the forcing term depend on the local solids concentration a semi discrete discontinuous finite volume element dfve scheme is proposed for this model the numerical method is constructed on a baseline finite element family of linear discontinuous elements for the approximation of velocity components and concentration field whereas the pressure is approximated by piecewise constant elements the unique solvability of both the nonlinear continuous problem and the semi discrete dfve scheme is discussed and optimal convergence estimates in several spatial norms are derived properties of the model and the predicted space accuracy of the proposed formulation are illustrated by detailed numerical examples including flows under gravity with changing direction a secondary settling tank in an axisymmetric setting and batch sedimentation in a tilted cylindrical vessel

finite volume methods for convection diffusion problems derivation stability and error analysis in both discrete h 1 and l 2 norms for cell centered finite volume approximations of convection diffusion problems are presented various upwind strategies are investigated the theoretical results are illustrated by numerical examples

unified analysis of finite volume methods for the stokes equations we establish a general framework for analyzing the class of finite volume methods for the stokes equations under the framework optimal order convergence for velocity and pressure in various sobolev norms are obtained in a natural and systematic way specifically optimal l 2 error estimates for velocity are obtained for the first time for several different finite volume methods

on the accuracy of the finite volume element method for diffusion equations on composite grids this paper develops the finite volume element fve method which is similar to the so called control volume finite element method but tailored to composite grid applications fve is described for the general two dimensional diffusion equation in divergence form o h 3 2 discretization error estimates are then developed for the case of triangular elements these results are improved to o h 2 using a special modification involving rectilinear elements at the grid interface

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

a discontinuous finite volume method for the stokes problems we develop a new discontinuous finite volume method for solving the stokes equations on both triangular and rectangular meshes an optimal error estimate for the approximation of velocity is obtained in a mesh dependent norm first order l 2 error estimates are derived for the approximations of both velocity and pressure

a finite volume method for stokes problems on quadrilateral meshes abstract we present a finite volume method for stokes problems using the isoparametric q 1 q 0 element pair on quadrilateral meshes to offset the lack of the i n f s u p condition a jump term of discrete pressure stabilizing term is added to the continuity approximation equation thus we establish a stabilized finite volume scheme on quadrilateral meshes then based on some superclose estimates we derive the optimal error estimates in the h 1 and l 2 norms for velocity and in the l 2 norm for pressure respectively numerical examples are provided to illustrate our theoretical analysis we emphasize that our work is the first time to propose and analyze a finite volume method for stoke problems using isoparametric elements on quadrilateral meshes

a finite volume method for solving navier stokes problems abstract we develop a finite volume method for solving the navier stokes equations on a triangular mesh we prove that the unique solution of the finite volume method converges to the true solution with optimal order for velocity and for pressure in discrete h 1 norm and l 2 norm respectively

weighted sum rate maximization for mimo broadcast channels using dirty paper coding and zero forcing methods we consider precoder design for maximizing the weighted sum rate wsr of successive zero forcing dirty paper coding szf dpc for this problem the existing precoder designs often assume a sum power constraint spc and rely on the singular value decomposition svd the svd based designs are known to be optimal but require high complexity we first propose a low complexity optimal precoder design for szf dpc under spc using the qr decomposition then we propose an efficient numerical algorithm to find the optimal precoders subject to per antenna power constraints papcs to this end the precoder design for papcs is formulated as an optimization problem with a rank constraint on the covariance matrices a well known approach to solve this problem is to relax the rank constraints and solve the relaxed problem interestingly for szf dpc we are able to prove that the rank relaxation is tight consequently the optimal precoder design for papcs is computed by solving the relaxed problem for which we propose a customized interior point method that exhibits a superlinear convergence rate two suboptimal precoder designs are also presented and compared to the optimal ones we also show that the proposed numerical method is applicable for finding the optimal precoders for block diagonalization scheme

a new discontinuous finite volume method for elliptic problems we develop and analyze a new discontinuous finite volume method for second order elliptic problems with tensor coefficients an optimal order error estimate is obtained in a mesh dependent norm an l2 error estimate is also obtained

a general mixed covolume framework for constructing conservative schemes for elliptic problems we present a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form i e written as first order systems we connect these schemes to standard mixed finite element methods via a one to one transfer operator between trial and test spaces in the nonsymmetric case convection diffusion equation we show one half order convergence rate for the flux variable which is approximated either by the lowest order raviart thomas space or by its image in the space of discontinuous piecewise constants in the symmetric case diffusion equation a first order convergence rate is obtained for both the state variable e g concentration and its flux numerical experiments are included

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 h 1 h 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

simplified weak galerkin and new finite difference schemes for the stokes equation abstract this article presents a simplified formulation for the weak galerkin finite element method for the stokes equations without using the degrees of freedom associated with the unknowns in the interior of each element as formulated in the original weak galerkin algorithm the simplified formulation preserves the important mass conservation property locally on each element and allows the use of general polygonal partitions a particular application of the simplified weak galerkin on rectangular partitions yields a new class of 5 and 7 point finite difference schemes for the stokes equations an explicit formula is presented for the computation of the element stiffness matrices on arbitrary polygonal elements error estimates of optimal order are established for the simplified weak galerkin finite element method in the h 1 and l 2 norms furthermore a superconvergence of order o h 3 2 is established on rectangular partitions for the velocity approximation in the h 1 norm at cell centers and a similar superconvergence is derived for the pressure approximation in the l 2 norm at cell centers some numerical results are reported to confirm the convergence and superconvergence theory