the numerical solution of the exterior dirichlet problem for helmholtz s equation via modified green s functions approach abstract in the 1970s a modified green s function approach for solving the helmholtz equation was proposed by jones and ursell and in the 1980s was clarified by kleinman roach and kress but to date there are no numerical results available for this approach in this paper a global galerkin method is used to numerically solve the exterior dirichlet problem for the helmholtz equation in three dimensions based on jones modified integral equation approach theoretical and computational details of the method are presented

a priori error estimates for interior penalty discontinuous galerkin method applied to nonlinear sobolev equations abstract a discontinuous galerkin method with interior penalties is presented for nonlinear sobolev equations a semi discrete and a family of fully discrete time approximate scheme are formulated these schemes can be symmetric or nonsymmetric hp version error estimates are analyzed for these schemes just because of a damping term b u u t included in sobolev equation which is the distinct character different from parabolic equation special test functions are chosen to deal with this term finally a priori l h 1 error estimate is derived for the semi discrete time scheme and similarly l h 1 and l 2 h 1 for the fully discrete time schemes these results also indicate that spatial rates in h 1 and time truncation errors in l 2 are optimal

discontinuous galerkin methods for incompressible and nearly incompressible elasticity by nitsche s method we propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes we show optimal error estimates that are uniform with respect to poisson s ratio the method is thus locking free we also introduce an equivalent mixed formulation allowing for completely incompressible elasticity problems numerical results are presented

piecewise polynomial collocation for boundary integral equations this paper considers the numerical solution of boundary integral equations of the second kind for laplace s equation delta u 0 on connected regions d in bf r 3 with boundary s the boundary s is allowed to be smooth or piecewise smooth and we let delta k 1 leq k leq n be a triangulation of s the numerical method is collocation with approximations which are piecewise quadratic in the parametrization variables leading to a numerical solution u n superconvergence results for u n are given for s a smooth surface and for a special type of refinement strategy for the triangulation we show that u u n is o delta 4 log delta at the collocation node points with delta being the mesh size for delta k error analyses are given are given for other quantities and an important error analysis is given for the approximation of s by piecewise quadratic interpolation on each triangular element with s either smooth or piecewise smooth the convergence result we

the efficient rotational pressure correction schemes for the coupling stokes darcy problem abstract in this paper the rotational pressure correction methods for the stokes darcy system are developed and analyzed the central advantage of these methods is a time dependent version of domain decomposition these methods have first order second order accuracy without the incompressibility constraint of the stokes darcy system their main feature is the implementation efficiency in that we only solve one vector valued elliptic equation and one scalar valued poisson equation for the stokes equations per time step the unconditional stability and long time stability are established and numerical experiments are also presented to show their performance

fast collocation methods for second kind integral equations in this paper we develop fast collocation methods for integral equations of the second kind with weakly singular kernels for this purpose we construct multiscale interpolating functions and collocation functionals having vanishing moments moreover we propose a truncation strategy for the coefficient matrix of the corresponding discrete system which forms a basis for fast algorithms an optimal order of convergence of the approximate solutions obtained from the fast algorithms is proved and the computational complexity of the algorithms is estimated the stability of the numerical method and the condition number of the truncated coefficient matrix are analyzed

coupling fluid flow with porous media flow 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

boundary knot method for some inverse problems associated with the helmholtz equation the boundary knot method is an inherently meshless integration free boundary type radial basis function collocation technique for the solution of partial differential equations in this paper the method is applied to the solution of some inverse problems for the helmholtz equation including the highly ill posed cauchy problem since the resulting matrix equation is badly ill conditioned a regularized solution is obtained by employing truncated singular value decomposition while the regularization parameter for the regularization method is provided by the l curve method numerical results are presented for both smooth and piecewise smooth geometry the stability of the method with respect to the noise in the data is investigated by using simulated noisy data the results show that the method is highly accurate computationally efficient and stable and can be a competitive alternative to existing methods for the numerical solution of the problems copyright 2005 john wiley sons ltd

adaptive stabilization of discontinuous galerkin methods for nonlinear elasticity motivation formulation and numerical examples the goal of this paper is to motivate introduce and demonstrate a novel approach to stabilizing discontinuous galerkin dg methods in nonlinear elasticity problems the stabilization term adapts to the solution of the problem by locally changing the size of a penalty term on the appearance of discontinuities with the goal of better approximating the solution consequently it is called an adaptive stabilization strategy the need for such a strategy is motivated through two and three dimensional examples in nonlinear elasticity the proposed scheme is simple to implement and compute and its performance is demonstrated with two and three dimensional numerical examples the accuracy of the proposed method is compared against a conforming method of the same order and a dg method with a traditional form of stabilization results for trilinear hexahedral elements indicate that the new stabilization strategy is more robust and more accurate when compared to a traditional form of stabilization however conforming trilinear hexahedral elements proved to be more computationally efficient for the examples shown here a two dimensional example with linear triangular elements showed comparable performances between the proposed method and a conforming one

error analysis for optimal control problem governed by convection diffusion equations dg method in the current paper we study the convergence properties of the dgfe approximation of optimal control problem governed by convection diffusion equations we derive a posteriori error estimates and a priori error estimates for both the states ad joint and the control variable approximation for the optimal control problem these estimates are apparently not available in the literature

rigorous sensitivity analysis for parameter dependent systems of equations abstract in this paper we construct rigorous and realistic enclosures for the solutions of parameter dependent systems of equations f x t 0 the enclosure of the solution x x t is given in the form x 0 s t t 0 with interval vectors x0 t0 and an interval matrix s this allows a detailed analysis of the sensitivity of the solution x t to small changes of t and the influence of specific changes in selected parameters on the solution the linear case is treated separately

a multiscale discontinuous galerkin method for convection diffusion reaction problems we provide a general framework of multiscale discontinuous galerkin methods developed in buffa et al 2006 hughes et al 2006 for general second order partial differential equations we establish stability of the method and prove the error estimates

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

a weak galerkin finite element method for a coupled stokes darcy problem on general meshes abstract in this paper we introduce and analyze a weak galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow flows are governed by the stokes equations in primal velocity pressure formulation and darcy equation in the second order primary formulation respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law by using the weak galerkin approach we consider the two dimensional problem with the usual polynomials of degree k 1 for the velocity and hydraulic head while polynomials of degree k 1 for the pressure the velocity and hydraulic head is enhanced by polynomials of degree k 1 on the edge of a finite element partition this new method has a lot of attractive computational features more general finite element partitions of arbitrary polygons with certain shape regularity fewer degrees of freedom and parameter free stability and error estimates of optimal order are obtained moreover numerical experiences are presented to illustrate the good performance confirm the optimal order of convergence and verify the efficiency of the proposed weak galerkin method in this paper we also deal with finite element partitions consisting of general meshes such as triangular mesh quadrilateral mesh hexagonal dominant mesh and voronoi mesh for the numerical weak galerkin approximation

a domain decomposition method for the steady state navier stokes darcy model with beavers joseph interface condition this paper proposes and analyzes a robin type multiphysics domain decomposition method ddm for the steady state navier stokes darcy model with three interface conditions in addition to the two regular interface conditions for the mass conservation and the force balance the beavers joseph condition is used as the interface condition in the tangential direction the major mathematical difficulty in adopting the beavers joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system y cao et al comm math sci 8 2010 pp 1 25 y cao et al siam j numer anal 47 2010 pp 4239 4256 in this paper the well posedness of the navier stokes darcy model with beavers joseph condition is analyzed by using a branch of nonsingular solutions by following the idea in y cao et al numer math 117 2011 pp 601 629 the three physical interface conditions are utilized together to construct the robin type boundary conditions on th

a one domain approach for modeling and simulation of free fluid over a porous medium we propose a one domain approach based on the brinkman model for the modeling and simulation of the transport phenomenon between free fluid and a porous medium a thin transition layer is introduced between the free fluid region and the porous media region across which the porosity and permeability undergo a rapid but continuous change we study the behavior of the solution to the one domain model analytically and numerically using the method of matched asymptotic expansion we recover the beavers joseph saffman bjs interface condition as the thickness of the transition layer goes to zero we also calculate the error estimates between the leading order solution of the one domain model and the standard darcy stokes model of two domain model with bjs condition numerical methods are developed for both the one domain model and the two domain model numerical results are presented to support the analytical results thereby justifying the one domain model as a good approximation to the two domain stokes darcy model

a strongly conservative finite element method for the coupled stokes biot model abstract in this paper based on interior penalty discontinuous galerkin method and mixed finite element method we consider a strongly conservative discretization for the rearranged stokes biot model theoretical analysis and numerical modeling are carried out from the view of mathematics we firstly present the existence and uniqueness of solution of the numerical scheme then the analysis of stability and priori error estimates are derived from the point of numerical simulation we report the numerical examples under uniform meshes which well validate the analysis of convergence and the strong mass conservation

rigorous sensitivity analysis for systems of linear and nonlinear equations methods are presented for performing a rigorous sensitivity analysis of numerical problems with independent noncorrelated data for general systems of linear and nonlinear equations the methods may serve for the following two purposes first to bound the dependency of the solution on changes in the input data in contrast to condition numbers a componentwise sensitivity analysis of the solution vector is performed second to estimate the true solution set for problems the input data of which are afflicted with tolerances the methods presented are very effective with the additional property that due to an automatic error control mechanism every computed result is guaranteed to be correct examples are given for linear systems demonstrating that the computed bounds are in general very sharp interesting comparisons to traditional condition numbers are given

a note on wave number dependence of wavelet matrix compression for integral equations with oscillatory kernel this paper analyzes the effect of large wave numbers on the wavelet method for integral equations arising in electromagnetic applications it is shown that the compression of the stiffness matrix deteriorates with increasing wave number a characteristic that has been reported before in the literature here however the exact dependence on the wave number is calculated analytically for the two dimensional helmholtz problem

mortar multiscale finite element methods for stokes darcy flows we investigate mortar multiscale numerical methods for coupled stokes and darcy flows with the beavers joseph saffman interface condition the domain is decomposed into a series of subdomains coarse grid of either stokes or darcy type the subdomains are discretized by appropriate stokes or darcy finite elements the solution is resolved locally in each coarse element on a fine scale allowing for non matching grids across subdomain interfaces coarse scale mortar finite elements are introduced on the interfaces to approximate the normal stress and impose weakly continuity of the normal velocity stability and a priori error estimates in terms of the fine subdomain scale h h and the coarse mortar scale h h are established for fairly general grid configurations assuming that the mortar space satisfies a certain inf sup condition several examples of such spaces in two and three dimensions are given numerical experiments are presented in confirmation of the theory

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

fast verification of solutions of matrix equations in this paper we are concerned with a matrix equation ax b where a is an n times n real matrix and x and b are n vectors assume that an approximate solution tilde x is given together with an approximate lu decomposition we will present fast algorithms for proving nonsingularity of a and for calculating rigorous error bounds for a 1 b tilde x infty the emphasis is on rigour of the bounds the purpose of this paper is to propose different algorithms the fastest with frac 2 3 n 3 flops computational cost for the verification step the same as for the lu decomposition the presented algorithms exclusively use library routines for lu decomposition and for all other matrix and vector operations

new finite element methods in computational fluid dynamics by h div elements in this paper the authors present two formulations for the stokes problem which make use of the existing h rm div elements of the raviart thomas type originally developed for the second order elliptic problems in addition two new h rm div elements are constructed and analyzed particularly for the new formulations optimal order error estimates are established for the corresponding finite element solutions in various sobolev norms the finite element solutions feature a full satisfaction of the continuity equation when existing raviart thomas type elements are employed in the numerical scheme

domain decomposition for linear elasticity with dg jumps and mortars in this article we present a domain decomposition method for solving a linear elasticity system data are transmitted by jumps as in a discontinuous galerkin method and mortars are introduced at the interfaces to dissociate the computation between neighboring subdomains a decoupling algorithm condenses the unknowns on the interface the matrix of the system is constructed by parallel computation numerical experiments confirming the theory are presented

an hp local discontinuous galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type in this paper an hp local discontinuous galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type on hp quasiuniform meshes using the brouwer fixed point theorem it is shown that the discrete problem has a solution and then using lipschitz continuity of the discrete solution map uniqueness is also proved a priori error estimates in broken h 1 norm and l 2 norm which are optimal in h suboptimal in p are derived these results are exactly the same as in the case of linear elliptic boundary value problems numerical experiments are provided to illustrate the theoretical results

a unified stabilized method for stokes and darcy s equations we use the lowest possible approximation order piecewise linear continuous velocities and piecewise constant pressures to compute solutions to stokes equation and darcy s equation applying an edge stabilization term to avoid locking we prove that the formulation satisfies the discrete inf sup condition we prove an optimal a priori error estimate for both problems the formulation is then extended to the coupled case using a nitsche type weak formulation allowing for different meshes in the two subdomains finally we present some numerical examples verifying the theoretical predictions and showing the flexibility of the coupled approach

on the solution of coupled stokes darcy model with beavers joseph interface condition abstract in this paper we first get the uniqueness result of the possible solution to the steady state coupled stokes darcy model with beavers joseph interface condition for any physical parameters especially for any 0 then we show the existence of solutions for any 0 by using galerkin method furthermore we analyze the error of the corresponding coupled finite element scheme and derive the optimal error estimates

geometric multigrid for darcy and brinkman models of flows in highly heterogeneous porous media we apply geometric multigrid methods for the finite element approximation of flow problems governed by darcy and brinkman systems used in modeling highly heterogeneous porous media the method is based on divergence conforming discontinuous galerkin methods and overlapping patch based domain decomposition smoothers we show in benchmark experiments that the method is robust with respect to mesh size and contrast of permeability for highly heterogeneous media

boundary integral equation method for source localization with a continuous wave sonar in this paper matched field processing is combined with the boundary integral equation method biem of scattering theory to study a sound source localization problem in a perturbed shallow ocean it is assumed that there is a known inclusion embedded in a shallow water waveguide continuous waves cw produced by a sound source are scattered by the inclusion and then received by a hydrophone array because the symmetry of the waveguide has been destroyed by the existence of the inclusion a proper procedure is required to avoid the mismatching a numerical scheme is presented that makes use of the separation of the source and the detection array and greatly reduces the computation a numerical simulation using this method is presented

analysis of hp discontinuous galerkin methods for incompressible two phase flow in this paper we prove the convergence of a class of discontinuous galerkin methods for solving the fully coupled incompressible two phase flow problem in the non degenerate case estimates in both the mesh size and the polynomial degrees are obtained numerical convergence rates confirm the theoretical results

a posteriori error estimates of discontinuous galerkin methods for the signorini problem a reliable and efficient a posteriori error estimator is derived for a class of discontinuous galerkin dg methods for the signorini problem a common property shared by many dg methods leads to a unified error analysis with the help of a constraint preserving enriching map the error estimator of dg methods is comparable with the error estimator of the conforming methods numerical experiments illustrate the performance of the error estimator

a hybrid high order method for a coupled stokes darcy problem on general meshes abstract in this work a hybrid high order hho method on general meshes is presented to solve a coupled stokes darcy problem with the beavers joseph saffman interface condition constructed on polynomials of arbitrary degree k 0 the numerical method is established in terms of discrete unknowns attached to mesh faces and cells or elements the unified discrete scheme for stokes equation and darcy equation is given by the continuity condition of the interface the unique solvability of the discrete scheme is proved moreover the energy error estimate for the velocity and l 2 error estimate for pressure of order k 1 are derived finally a series of numerical experiments are reported to illustrate the accuracy mass conservation and robustness of our method

solving the advection diffusion equation on unstructured meshes with discontinuous mixed finite elements and a local time stepping procedure explicit schemes are known to provide less numerical diffusion in solving the advection diffusion equation especially for advection dominated problems traditional explicit schemes use fixed time steps restricted by the global cfl condition in order to guarantee stability this is known to slow down the computation especially for heterogeneous domains and or unstructured meshes to avoid this problem local time stepping procedures where the time step is allowed to vary spatially in order to satisfy a local cfl condition have been developed in this paper a local time stepping approach is used with a numerical model based on discontinuous galerkin mixed finite element methods to solve the advection diffusion equation the developments are detailed for general unstructured triangular meshes numerical experiments are performed to show the efficiency of the numerical model for the simulation of i the transport of a solute on highly unstructured meshes and ii density driven flow where the velocity field changes at each time step the model gives stable results with significant reduction of the computational cost especially for the non linear problem moreover numerical diffusion is also reduced for highly advective problems copyright 2009 john wiley sons ltd

modified error bounds for approximate solutions of dense linear systems abstract we derive verified error bounds for approximate solutions of dense linear systems there are verification methods using an approximate inverse of a coefficient matrix as a preconditioner where the preconditioned coefficient matrix is likely to be anh matrix also known as a generalized diagonally dominant matrix we focus on two inclusion methods of matrix multiplication for the preconditioning and propose verified error bounds adapted to the inclusion methods these proposed error bounds are tighter than conventional ones especially in critically ill conditioned cases numerical results are presented showing the effectiveness of the proposed error bounds

a local projection operator for quadrilateral finite elements this note studies the approximation error of a local projection operator on polynomials of total degree k defined on quadrilaterals among other applications this projection operator permits to derive easily error estimates for quadrature formulas

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

a priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems we analyze three discontinuous galerkin approximations for solving elliptic problems in two or three dimensions in each one the basic bilinear form is nonsymmetric the first one has a penalty term on edges the second has one constraint per edge and the third is totally unconstrained for each of them we prove hp error estimates in the h1 norm optimal with respect to h the mesh size and nearly optimal with respect to p the degree of polynomial approximation we establish these results for general elements in two and three dimensions for the unconstrained method we establish a new approximation result valid on simplicial elements l2 estimates are also derived for the three methods

trigonometric wavelets for hermite interpolation the aim of this paper is to investigate a multiresolution analysis of nested subspaces of trigonometric polynomials the pair of scaling functions which span the sample spaces are fundamental functions for hermite interpolation on a dyadic partition of nodes on the interval 0 2 two wavelet functions that generate the corresponding orthogonal complementary subspaces are constructed so as to possess the same fundamental interpolatory properties as the scaling functions together with the corresponding dual functions these interpolatory properties of the scaling functions and wavelets are used to formulate the specific decomposition and reconstruction sequences consequently this trigonometric multiresolution analysis allows a completely explicit algorithmic treatment

coupling discontinuous galerkin and mixed finite element discretizations using mortar finite elements discontinuous galerkin dg and mixed finite element mfe methods are two popular methods that possess local mass conservation in this paper we investigate dg dg and dg mfe domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface the subdomain grids need not match and the mortar grid may be much coarser giving a two scale method convergence results in terms of the fine subdomain scale h and the coarse mortar scale h are established for both types of couplings in addition a nonoverlapping parallel domain decomposition algorithm is developed which reduces the coupled system to an interface mortar problem the properties of the interface operator are analyzed

on the solution of the coupled navier stokes and darcy equations abstract this paper introduces and analyzes two mathematical models for coupling the incompressible navier stokes equations with the porous media flow equations a numerical method that uses continuous finite elements in the incompressible flow region and discontinuous finite elements in the porous medium is proposed existence and uniqueness results under small data condition of the numerical solution are proved optimal a priori error estimates are derived numerical examples comparing the two models under varying physical parameters are provided

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

a divergence free weak virtual element method for the stokes darcy problem on general meshes abstract this paper presents a weak virtual element method on general meshes for the stokes darcy problem with the beavers joseph saffman interface condition the velocity is discretized by the h div virtual element the pressure is approximated by discontinuous piecewise polynomials besides a polynomial space on the element faces is introduced to approximate the tangential trace of the velocity in the stokes equations the velocity on the discrete level is exactly divergence free and thus the exact mass conservation is preserved in the discretization the well posedness of the discrete problem is proved and an a priori error estimate is derived that implies the error for the velocity in a suitable norm does not depend on the pressure a series of numerical experiments are reported to illustrate the performance of the method

a fast collocation method for acoustic scattering in shallow oceans abstract in this paper we investigate the numerical solution of the integral equation of the second kind reduced by acoustic scattering in shallow oceans with dirichlet condition based on analyzing the singularity of the truncating kernel with a sum of infinite series using our trigonometric interpolatory wavelets and collocation method we obtain the numerical solution which possesses a fast convergence rate like o 2 j moreover the entries of the stiffness matrix can be obtained by fft which lead the computational complexity to decrease obviously

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

overestimation in linear interval equations bounds are given for the overestimation of the solution set of a system of linear interval equations by various inclusion intervals for this set in particular a theorem of gay this journal 19 1982 pp 858 870 on the quadratic approximation property of the preconditioned fixpoint inverse is strengthened

a comparison of some methods for solving linear interval equations certain cases in which the interval hull of a system of linear interval equations can be computed inexpensively are outlined we extend a proposed technique of hansen and rohn with a formula that bounds the solution set of a system of equations whose coefficient matrix bf a underline a overline a is an h matrix when bf a is centered about a diagonal matrix these bounds are the smallest possible i e the bounds are then the solution hull hansen s scheme also computes the solution hull when the linear interval system bf a bf x bf b underline b overline b is such that bf a is inverse positive and underline b overline b neq 0 earlier results of others also imply that when bf a is an m matrix and bf b geq 0 bf b leq 0 or 0 in bf b interval gaussian elimination iga computes the hull we also give a method of computing the solution hull inexpensively in many instances when bf a is inverse positive given an outer approximation such as that obtained from iga examples are used to compare these schemes under various conditions

a posteriori error estimate for the h d i v conforming mixed finite element for the coupled darcy stokes system an h d i v conforming mixed finite element method has been proposed for the coupled darcy stokes flow in kanschat and riviere 2010 12 which imposes normal continuity on the velocity field strongly across the darcy stokes interface here we develop an a posteriori error estimator for this h d i v conforming mixed method and prove its global reliability and efficiency due to the strong coupling on the interface special techniques need to be employed in the proof this is the main difference between this paper and babuska and gatica s 2010 work 38 in which they analyzed an a posteriori error estimator for the mixed formulation using weakly coupled interface conditions

a stabilized finite volume method for the evolutionary stokes darcy system abstract in this paper we propose a stabilized fully discrete finite volume method based on two local gauss integrals for a non stationary stokes darcy problem this stabilized method is free of stabilized parameters and uses the lowest equal order finite element triples p 1 p 1 p 1 for approximating the velocity pressure and hydraulic head of the stokes darcy model under a modest time step restriction in relation to physical parameters we give the stability analysis and the error estimates for the stabilized finite volume scheme by means of a relationship between finite volume and finite element approximations with the lower order elements finally a series of numerical experiments are provided to demonstrate the validity of the theoretical results

an augmented cartesian grid method for stokes darcy fluid structure interactions summary a new finite difference method based on cartesian meshes is proposed for solving the fluid structure interaction between a fluid flow modeled by the stokes equations and a porous media modeled by the darcy s law the idea is to introduce several augmented variables along the interface between the fluid flow and the porous media so that the problem can be decoupled as several poisson equations the augmented variables should be chosen so that the beavers joseph saffman and other interface conditions are satisfied in the discretization the augmented variables have co dimension one compared with that of the primitive variables and are solved through the schur complement system a non trivial analytic solution with a circular interface is constructed to check second order convergency of the proposed method numerical examples with various interfaces and parameters are also presented some simulations show interesting behaviors of the fluid structure interaction between the fluid flow and the porous media the computational framework can be applied to other multi phase and multi physics problems copyright 2015 john wiley sons ltd

robin robin domain decomposition methods for the steady state stokes darcy system with the beavers joseph interface condition domain decomposition methods for solving the coupled stokes darcy system with the beavers joseph interface condition are proposed and analyzed robin boundary conditions are used to decouple the stokes and darcy parts of the system then parallel and serial domain decomposition methods are constructed based on the two decoupled sub problems convergence of the two methods is demonstrated and the results of computational experiments are presented to illustrate the convergence

interior penalty discontinuous approximations of convection diffusion problems with parabolic layers a nonsymmetric discontinuous galerkin finite element method with interior penalties is considered for two dimensional convection diffusion problems with regular and parabolic layers on an anisotropic shishkin type mesh with bilinear elements we prove error estimates uniformly in the perturbation parameter in an integral norm associated with this method on different types of interelement edges we derive the values of discontinuity penalization parameters numerical experiments complement the theoretical results

a posteriori error analysis of the fully discretized time dependent coupled darcy and stokes equations abstract we present an a posteriori error analysis of the fully discretized time dependent darcy and stokes equations that models laminar fluid flow over a porous medium in two or three dimensional connected open domains which are coupled via appropriate matching conditions on the interface the problem is discretized by the backward euler scheme in time and finite elements in space

a structure preserving discontinuous galerkin scheme for the fisher kpp equation an implicit euler discontinuous galerkin scheme for the fisher kolmogorov petrovsky piscounov fisher kpp equation for population densities with no flux boundary conditions is suggested and analyzed using an exponential variable transformation the numerical scheme automatically preserves the positivity of the discrete solution a discrete entropy inequality is derived and the exponential time decay of the discrete density to the stable steady state in the l 1 norm is proved if the initial entropy is smaller than the measure of the domain the discrete solution is proved to converge in the l 2 norm to the unique strong solution to the time discrete fisher kpp equation as the mesh size tends to zero numerical experiments in one space dimension illustrate the theoretical results

numerical analysis of two grad div stabilization methods for the time dependent stokes darcy model abstract we aim to present two algorithms for the non stationary stokes darcy model the first one is the standard grad div stabilization scheme the other one is a modular grad div based on the standard backward euler code which does not crash or slow down for large value grad div parameters both algorithms cannot only improve the efficiency and accuracy of calculation but also can improve mass conservation while the modular algorithm can be better we give a complete theoretical analysis of the stability and error estimations of the algorithms finally the theoretical results are verified by numerical experiments and the advantages of adding grad div stabilization terms are demonstrated

a strongly conservative finite element method for the coupling of stokes and darcy flow we consider a model of coupled free and porous media flow governed by stokes and darcy equations with the beavers joseph saffman interface condition this model is discretized using divergence conforming finite elements for the velocities in the whole domain discontinuous galerkin techniques and mixed methods are used in the stokes and darcy subdomains respectively this discretization is strongly conservative in h d i v w and we show convergence numerical results validate our findings and indicate optimal convergence orders

stencil reduction algorithms for the local discontinuous galerkin method the problem of reducing the stencil of the local discontinuous galerkin method applied to second order differential operator is discussed heuristic algorithms to minimize the total number of non zero blocks of the reduced stiffness matrix are presented and tested on a wide variety of unstructured and structured grids in 2d and 3d copyright 2009 john wiley sons ltd

a conforming mixed finite element method for the navier stokes darcy coupled problem in this paper we develop the a priori analysis of a mixed finite element method for the coupling of fluid flow with porous media flow flows are governed by the navier stokes and darcy equations respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law we consider the standard mixed formulation in the navier stokes domain and the dual mixed one in the darcy region which yields the introduction of the trace of the porous medium pressure as a suitable lagrange multiplier the finite element subspaces defining the discrete formulation employ bernardi raugel and raviart thomas elements for the velocities piecewise constants for the pressures and continuous piecewise linear elements for the lagrange multiplier we show stability convergence and a priori error estimates for the associated galerkin scheme finally several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are reported

accurate solution of dense linear systems part ii algorithms using directed rounding in part i and this part ii of our paper we investigate how extra precise evaluation of dot products can be used to solve ill conditioned linear systems rigorously and accurately in part i only rounding to nearest is used in this part ii we improve the results significantly by permitting directed rounding linear systems with tolerances in the data are treated and a comfortable way is described to compute error bounds for extremely ill conditioned linear systems with condition numbers up to about u 2 n where u denotes the relative rounding error unit in a given working precision we improve a method by hansen bliek rohn ning kearfott neumaier of the known methods by krawczyk rump hansen et al ogita and nguyen we show that our presented algorithm lsserrbnd seems the best compromise between accuracy and speed moreover for input data with tolerances a new method to compute componentwise inner bounds is presented for not too wide input data they demonstrate that the computed inclusions are often almost optimal all algorithms are given in executable matlab code and are available from my homepage

on l2 error estimate for nonsymmetric interior penalty galerkin approximation to linear elliptic problems with nonhomogeneous dirichlet data in this paper we present improved a priori error estimates for a nonsymmetric interior penalty galerkin method nipg with super penalty for the problem email protected f in w and u g on email protected using piecewise polynomials of degree less than or equal to r our new l 2 error estimate is of order h r r 1 2 when email protected h r 1 2 email protected and is optimal i e of order h r r 1 when email protected h r 1 email protected where h denotes the mesh size numerical experiments are presented to illustrate the theoretical results

a priori error estimates of a combined mixed finite element and local discontinuous galerkin method for an incompressible miscible displacement problem abstract a numerical approximation for a kind of incompressible miscible displacement problems in high dimension in porous media is studied mixed finite element method is applied to the flow equation and the transport one is solved by the local discontinuous galerkin method ldg based on interpolation projection properties and the induction hypothesis a priori hp error estimates are obtained numerical results are presented which verify the theoretical results

a geometric optics method for high frequency electromagnetic fields computations near fold caustics part ii the energy we present the computation of the amplitudes needed to evaluate the energy deposited by the laser wave in a plasma when a fold caustic forms we first recall the eulerian method designed in benamou et al j comput appl math 156 2003 93 to compute the caustic location and the phases associated to the two ray branches on its illuminated side we then turn to the computation of the amplitudes needed to evaluate the energy we use the classical geometrical form of the amplitudes to avoid the blow up problem at the caustic as our proposed method is eulerian we have to consider transport equations for these geometrical quantities where the advection field depends on the ray flow the associated vector field structurally vanishes like the square root of the distance to the caustic when approaching the caustic this introduces an additional difficulty as traditional finite difference scheme do not retain their accuracy for such advection fields we propose a new scheme which remains of order 1 at the caustic and present a partial theoretical analysis as well as a numerical validation we also test the capability of our eulerian geometrical algorithm to produce numerical solution of the helmholtz equation and attempt to check their frequency asymptotic accuracy

dg approximation of coupled navier stokes and darcy equations by beaver joseph saffman interface condition in this work we couple the incompressible steady navier stokes equations with the darcy equations by means of the beaver joseph saffman s condition on the interface under suitable smallness conditions on the data we prove existence of a weak solution as well as some a priori estimates we establish local uniqueness when the data satisfy additional smallness restrictions then we propose a discontinuous galerkin scheme for discretizing the equations and do its numerical analysis

estimation of penalty parameters for symmetric interior penalty galerkin methods this paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty galerkin methods in particular we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements numerical examples in all dimensions and for different polynomial degrees are presented we show the numerical effects of loss of coercivity

locally conservative coupling of stokes and darcy flows a locally conservative numerical method for solving the coupled stokes and darcy flows problem is formulated and analyzed the approach employs the mixed finite element method for the darcy region and the discontinuous galerkin method for the stokes region a discrete inf sup condition and optimal error estimates are derived

analysis of a nonsymmetric discontinuous galerkin method for elliptic problems stability and energy error estimates in this paper we analyze a nonsymmetric discontinuous galerkin method for elliptic problems proposed by oden babuska and baumann our main results are a complete inf sup stability analysis and as a consequence error estimates in a mesh dependent energy norm allowing variable meshsize and order of polynomials the analysis is carried out in two spatial dimensions on an unstructured triangulation

quasi optimal a priori interface error bounds and a posteriori estimates for the interior penalty method in this work we show quasi optimal interface error estimates for solutions obtained by the symmetric interior penalty discontinuous galerkin method it is proved that the numerical solution restricted to an interface converges with order lvert ln h rvert h k 1 under suitable regularity requirements where the logarithmic factor is only present in the lowest order case i e k 1 for this case we also derive and analyze two a posteriori error estimators which demonstrate that the jump terms of the discrete fluxes are not essential to obtain local efficiency and reliability we support our analysis by numerical results and demonstrate that the interface approximation can be locally postprocessed to obtain discrete solutions of order h k 1 2 in the energy norm

numerical analysis of the unconditionally stable discontinuous galerkin schemes for the nonstationary conduction convection problem abstract this work aims at employing the discontinuous galerkin dg methods for solving the nonstationary conduction convection model which is motivated by advantages of dg methods the discontinuous element pair p k p k 1 p k in the space discretization and backward euler scheme in time are applied as well as an explicit upwind scheme based on picard iteration is introduced for the coupled term existence and uniqueness of the approximation solutions are proved along with preserving the unconditional stability of the numerical schemes the priori error estimates for velocity pressure and temperature are also derived numerical tests are provided to verify the theoretical analysis and to indicate the efficiency of the dg methods in simulating practical problems

discontinuous galerkin methods for mass transfer through semipermeable membranes a discontinuous galerkin dg method for the numerical solution of initial boundary value multicompartment partial differential equation models interconnected with interface conditions is presented and analyzed the study of interface problems is motivated by models of mass transfer of solutes through semipermeable membranes more specifically a model problem consisting of a system of semilinear parabolic advection diffusion reaction partial differential equations in each compartment equipped with respective initial and boundary conditions is considered nonlinear interface conditions modeling selective permeability congestion and partial reflection are applied to the compartment interfaces an interior penalty dg method is presented for this problem and it is analyzed in the space discrete setting the a priori analysis shows that the method yields optimal a priori bounds provided the exact solution is sufficiently smooth numerical experiments indicate agreement with the theoretical bounds and highlight the stability of the numerical method in the advection dominated regime