on the finite volume element method for general self adjoint elliptic problems the finite volume element method fve is a discretization technique for partial differential equations this paper develops discretization energy error estimates for general self adjoint elliptic boundary value problems with fve based on triangulations on which there exist linear finite element spaces and a very general type of control volumes covolumes the energy error estimates of this paper are also optimal but the restriction conditions for the covolumes given in r e bank and d j rose siam j numer anal 24 1987 pp 777 787 z q cai numer math 58 1991 pp 713 735 are removed the authors finally provide a counterexample to show that an expected l2 error estimate does not exist in the usual sense it is conjectured that the optimal order of u u h 0 omega should be o h for the general case

making the oslo algorithm more efficient the oslo algorithm is a general method for adding knots to a b spline curve or tensor product b spline surface the method provides a framework in computer aided geometric design for both manipulating and rendering spine curves and surfaces and is derived from properties of discrete b splines in this paper we prove that all discrete b splines which are nonzero at a particular point can in general be considered as lower order discrete b splines on a subset of the knots we also give necessary and sufficient conditions for a discrete b spline to be a continuous function of its parameters these results are used to improve on the original oslo algorithms

higher order finite volume methods ii inf sup condition and uniform local ellipticity the main purpose of this paper is to study the construction of higher order finite volume methods fvms of triangle meshes we investigate the relationship of the three theoretical notions crucial in the construction of fvms the uniform ellipticity of the family of its discrete bilinear forms its inf sup condition and its uniform local ellipticity both the uniform ellipticity of the family of the discrete bilinear forms and its inf sup condition guarantee the unique solvability of the fvm equations and the optimal error estimate of the approximate solution we characterize the uniform ellipticity in terms of the inf sup condition and a geometric condition on the bijective operator mapping from the trial space onto the test space involved in the construction of fvms we present a geometric interpretation of the inf sup condition moreover since the uniform local ellipticity is a convenient sufficient condition for the uniform ellipticity we further provide sufficient conditions and necessary conditions of the uniform local ellipticity

a node centered finite volume method for a fracture model on triangulations abstract in this paper a node centered finite volume method based on triangulations for a fracture model is presented in which we restrict the pressure to the linear finite element space while the velocity can be approximated by constant vectors element by element the numerical scheme is established just associated with the pressure to avoid the saddle point problem error estimates of o h accuracy for the discrete h1 semi norm and the discrete l2 norm of pressure p and the l2 2 norm of velocity u are developed on general triangulations under an additional assumption about essentially symmetric control volumes the error estimates for the pressure p can be improved to o h3 2 finally numerical experiments are carried out to verify the accuracy and convergence rates for the proposed finite volume scheme

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

the finite volume element methods for poisson equation based on adini s element in this paper we construct and analyze the finite volume element scheme based on adini s element for poisson equation choosing trial and test spaces as the adini s finite element space and the piecewise linear function space respectively optimal order error estimates in h 1 and l 2 norms are derived in addition the global and locally pointwise superconvergence estimates are also discussed lastly a numerical experiment is carried out which confirms some results of theoretical analysis

a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes abstract we construct a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes where the strict convexity restriction on the meshes is removed the main contributions of this paper include three aspects firstly the classical finite volume element c fve method is extended to severely distorted quadrilateral meshes even with concave cells by virtue of a new overlapping dual partition and a new gradient approximation in fact the choice of this dual partition and gradient approximation is also conducive to the construction of a monotone scheme secondly a new monotonicity correction is suggested based on which we obtain a monotone finite volume element m fve method the resulting m fve method still keeps the local conservation and is easy for implementation finally we analyze theoretically the truncation error and the monotonicity for this scheme besides the existence of a solution to this nonlinear scheme is proved by applying the brouwer s fixed point theorem numerical results demonstrate that the m fve method has the approximate second order accuracy and preserves well the positivity of the solution for both isotropic and anisotropic diffusion problems on severely distorted quadrilateral meshes

homogenization for fully nonlinear parabolic equations abstract this paper concerns the homogenization of fully nonlinear parabolic equations of the form t u e h t x t e 2 x e d 2 u e 0 in 0 t r n where the hamiltonian h t x x is periodic both in and in our aim is to establish sufficient conditions for the convergence as e 0 of u e to a solution u to the effective equation t u h t x d 2 u 0 in 0 t r n where the effective hamiltonian h is obtained by a parabolic equation called cell problem we shall prove that h inherits several properties of h we also consider the case that u e 0 x h x x e on r n we point out a sufficient condition for having u 0 x h x on r n with an effective initial datum h given by the asymptotic behaviour of the solution to the recession problem a parabolic cauchy problem related to 1 1

superconvergence by l2 projection for a stabilized finite volume method for the stationary navier stokes equations a superconvergence result is established for the stationary navier stokes equations by a stabilized finite volume method and l 2 projection on a coarse mesh like other results in the family of l 2 projection methods the superconvergence presented in this paper is based on some regularity assumption for the navier stokes problem and is applicable to the stabilized finite volume method with quasi uniform partitions

the locally stabilized finite volume method for incompressible flow a locally stabilized finite volume method for the two dimensional incompressible flow governed by stationary stokes and navier stokes equations is investigated in this work a macroelement condition is introduced for constructing the local stabilized formulation for the problem the optimal l 2 error estimate of u h p h determined by stokes equations is also deduced some numerical examples are provided to check the efficiency of this method

enriched multi point flux approximation for general grids it is well known that the two point flux approximation a numerical scheme used in most commercial reservoir simulators has o 1 error when grids are not k orthogonal in the last decade the multi point flux approximations have been developed as a remedy however non physical oscillations can appear when the anisotropy is really strong we found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation in this paper we propose the control volume enriched multi point flux approximation empfa for general diffusion problems on polygonal and polyhedral meshes non physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor exact linear solutions are recovered for grids with non planar interfaces and a first and second order convergence are achieved for the flux and scalar unknowns respectively

an integral equation approach to the unsteady convection diffusion equations an integral equation approach is proposed to solve the unsteady convection diffusion equations in this approach the second order adams moulton method is firstly utilized for the time discretization then by using the green s function of the laplace equation in the series form the convection diffusion equation is transformed into an integral equation that is further converted into an algebraic equation system the accuracy convergence and the stability of this integral equation approach are examined by four examples in comparison with the characteristic variational multiscale method and the finite volume element method the integral equation approach shows a higher accuracy compared with the finite volume element method the integral equation approach has a better convergence in solving the convection dominated convection diffusion problems the integral equation approach demonstrates a good stability

on domain decomposition algorithms for covolume methods for elliptic problems in this paper some new two level additive schwarz domain decomposition methods are constructed and used as preconditioners for gmres for the nonsymmetric systems arising from the covolume methods for elliptic problems without any restriction on the coarse mesh size the convergence of the preconditioned gmres is obtained for both circumcentric and barycentric covolume methods with inexact solvers for subproblems the rates of convergence are shown to be independent of the number of subregions and these methods are optimal for generous overlap

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

the method of variably scaled radial kernels for solving two dimensional magnetohydrodynamic mhd equations using two discretizations the crank nicolson scheme and the method of lines mol abstract mhd equations have many applications in physics and engineering the model is coupled equations in velocity and magnetic field and has a parameter namely hartmann the value of hartmann number plays an important role in the equations when this parameter increases using different meshless methods makes the oscillations in velocity near the boundary layers in the region of the problem in the present paper a numerical meshless method based on radial basis functions rbfs is provided to solve mhd equations for approximating the spatial variable a new approach which is introduced by bozzini et al 2015 is applied the method will be used here is based on the interpolation with variably scaled kernels the methodology of the new technique is defining the scale function c on the domain r d then the interpolation problem from the data locations x j r d transforms to the new interpolation problem in the data locations x j c x j r d 1 bozzini et al 2015 the radial kernels used in the current work are multiquadrics mq inverse quadric iq and wendland s function of course the latter one is based on compactly supported functions to discretize the time variable two techniques are applied one of them is the crank nicolson scheme and another one is based on mol the numerical simulations have been carried out on the square and elliptical ducts and the obtained numerical results show the ability of the new method for solving this problem also in appendix we provide a computational algorithm for implementing the new technique in matlab software

the finite volume method and application in combinations abstract in this paper the conventional finite volume method fvm is interpreted as a new kind of galerkin finite element method fem where the same piecewise linear functions are chosen as in both trial and test spaces and some specific integration rules are adopted error analysis is made for the regular delaunay triangulation involving obtuse triangles separated to prove optimal convergence rates of the approximate solutions obtained the new interpretation makes the fvm analysis much easier because we may bypass verification of the nontrivial ladyzhenskaya babuska brezzi lbb condition by the petrov galerkin fem in the existing analysis of fvm more importantly the new interpretation and the simple fvm analysis enable us to construct easily the combinations of fvm with other popular numerical methods such as the finite element method fem the finite difference method fdm the ritz galerkin method rgm etc for solving complicated problems of partial differential equations pde for example for solving singularity problems the combination of rgm fvm is superior to the combination of rgm fdm in flexibility of arbitrary solution domains and also superior to the combination of rgm fem in substantial saving of cpu time since the conservative law of flux may be maintained exactly in the numerical solutions and since obtuse triangles may be included in the delaunay triangulation the fvm and its combinations become very promising for solving elliptic boundary value problems in particular those where singularity solutions exist and those where the obeying of conservative law is crucial the numerical examples of combinations of rgm fvm are given for solving motz s problems to verify the optimal convergence rates the techniques and analysis of fvm and its combinations can be extended to the convection diffusion problems most importantly an important aspect in this paper is the possibility to include obtuse triangles in the error analysis

the finite volume method based on stabilized finite element for the stationary navier stokes problem a finite volume method based on stabilized finite element for the two dimensional stationary navier stokes equations is investigated in this work a macroelement condition is introduced for constructing the local stabilized formulation for the problem we obtain the well posedness of the fvm based on stabilized finite element for the stationary navier stokes equations moreover for quadrilateral and triangular partition the optimal h 1 error estimate of the finite volume solution u h and l 2 error estimate for p h are introduced finally we provide a numerical example to confirm the efficiency of the fvm

convergence of finite volume schemes for semilinear convection diffusion equations the topic of this work is the discretization of semilinear elliptic problems in two space dimensions by the cell centered finite volume method dirichlet boundary conditions are considered here a discrete poincare inequality is used and estimates on the approximate solutions are proven the convergence of the scheme without any assumption on the regularity of the exact solution is proven using some compactness results which are shown to hold for the approximate solutions

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

symmetric modified finite volume element methods for self adjoint elliptic and parabolic problems the finite volume element method is a discretization technique for partial differential equations but in general case the coefficient matrix of its linear system is not symmetric even for the self adjoint continuous problem in this paper we develop a kind of symmetric modified finite volume element methods both for general self adjoint elliptic and for parabolic problems on general discretization their coefficient matrix are symmetric we give the optimal order energy norm error estimates we also prove that the difference between the solutions of the finite volume element method and symmetric modified finite volume element method is a high order term

a symmetric finite volume scheme for selfadjoint elliptic problems based on a linear finite element space a symmetric finite volume scheme for a self adjoint elliptic boundary value problem is proposed error estimates in l2 norm h1 norm and l norm are derived some post processing techniques are also provided

an algorithm for multi resolution analysis of nurbs surfaces a new algorithm is proposed for the multi resolution representation of nurbs non uniform rational b spline surfaces with boundary consistency constraints the kernel idea is to employ t meshes t spline control meshes to construct the multi resolution representation for surfaces to this end the original b mesh b spline control mesh is first divided into a boundary part and an inner one then the multi resolution representation of the inner part is achieved by wavelet transform at last the topology of the boundary part is added onto the obtained multi resolution meshes to construct the final t spline based multi resolution representation result the boundary shapes keep unchanged in the whole process of the treatment because the boundary consistency constraints are always contained in the boundary part this advantage of the new algorithm is demonstrated by three examples

the mortar finite volume method with the crouzeix raviart element for elliptic problems abstract we construct and analyse a mortar finite volume method for the discretization for selfadjoint elliptic boundary value problems in r 2 this method is based on the mortar crouzeix raviart non conforming finite element spaces we prove the optimal order h 1 norm and l 2 norm error estimates between the exact solution and the mortar finite volume approximation of the elliptic problems

a low order anisotropic nonconforming characteristic finite element method for a convection dominated transport problem a low order anisotropic nonconforming rectangular finite element method for the convection diffusion problem with a modified characteristic finite element scheme is studied in this paper the o h 2 order error estimate in l 2 norm with respect to the space one order higher than the expanded characteristic mixed finite element scheme with order o h and the same as the conforming case for a modified characteristic finite element scheme under regular meshes is obtained by use of some distinct properties of the interpolation operator and the mean value technique instead of the so called elliptic projection which is an indispensable tool in the convergence analysis of the previous literature lastly some numerical results of the element are provided to verify our theoretical analysis

the averaged control system of fast oscillating control systems for control systems that either have a fast explicit periodic dependence on time and bounded controls or have periodic solutions and small controls we define an average control system that takes into account all possible variations of the control and prove that its solutions approximate all solutions of the oscillating system as the frequency of the oscillations tends to infinity the dimension of its velocity set is characterized geometrically when it is maximum the average system defines a finsler metric which is not twice differentiable in general for minimum time control this average system allows one to give a rigorous proof that averaging the hamiltonian given by the maximum principle is a valid approximation

an optimal weighted upwinding covolume method on non standard grids for convection diffusion problems in 2d in this paper we develop an optimal weighted upwinding covolume method on non standard covolume grids for convection diffusion problems in two dimensions the novel feature of our method is that we construct the non standard covolume grid in which the nodes of covolumes vary in the interior of different volumes of primary grid depending on the local weighted factors and further on the local peclet s numbers a simple method of finding the local optimal weighted factors is also derived from a non linear function of local peclet s numbers the developed method leads to a totally new scheme for convection diffusion problems which overcomes numerical oscillation avoids numerical dispersion and has high order accuracy some theoretical analyses are given and numerical experiments are presented to illustrate the performance of the method copyright 2006 john wiley sons ltd

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

lossless and near lossless image compression based on multiresolution analysis there are applications in data compression where quality control is of utmost importance certain features in the decoded signal must be exactly or very accurately recovered yet one would like to be as economical as possible with respect to storage and speed of computation in this paper we present a multi scale data compression algorithm within harten s interpolatory framework for multiresolution that gives a specific estimate of the precise error between the original and the decoded signal when measured in the l and in the l p p 1 2 discrete norms the proposed algorithm does not rely on a tensor product strategy to compress two dimensional signals and it provides a priori bounds of the peak absolute error pae the root mean square error rmse and the peak signal to noise ratio psnr of the decoded image that depend on the quantization parameters in addition after data compression by applying this non separable multi scale transformation the user has an the exact value of the pae rmse and psnr before the decoding process takes place we show how this technique can be used to obtain lossless and near lossless image compression algorithms

a finite volume method based on the crouzeix raviart element for elliptic pde s in two dimensions we introduce and analyse a finite volume method for the discretization of elliptic boundary value problems in mathbb r 2 the method is based on nonuniform triangulations with piecewise linear nonconforming spaces we prove optimal order error estimates in the l 2 norm and a mesh dependent h 1 norm

biquadratic finite volume element methods based on optimal stress points for parabolic problems in this paper the semi discrete and full discrete biquadratic finite volume element schemes based on optimal stress points for a class of parabolic problems are presented optimal order error estimates in h 1 and l 2 norms are derived in addition the superconvergences of numerical gradients at optimal stress points are also discussed a numerical experiment confirms some results of theoretical analysis

a multiresolution analysis for tensor product splines using weighted spline wavelets we construct biorthogonal spline wavelets for periodic splines which extend the notion of lazy wavelets for linear functions where the wavelets are simply a subset of the scaling functions to splines of higher degree we then use the lifting scheme in order to improve the approximation properties with respect to a norm induced by a weighted inner product with a piecewise constant weight function using the lifted wavelets we define a multiresolution analysis of tensor product spline functions and apply it to image compression of black and white images by performing as a model problem image compression with black and white images we demonstrate that the use of a weight function allows to adapt the norm to the specific problem

a finite volume method on general surfaces and its error estimates abstract in this paper we study a finite volume method and its error estimates for the numerical solution of some model second order elliptic partial differential equations defined on a smooth surface the discretization is defined via a surface mesh consisting of piecewise planar triangles and piecewise polygons the optimal error estimates of the approximate solution are proved in both the h 1 and l 2 norms which are of first order and second order respectively under mesh regularity assumptions some numerical tests are also carried out to experimentally verify our theoretical analysis

superconvergence of quadratic finite volume method on triangular meshes abstract this paper is concerned with the superconvergence properties of the quadratic finite volume method fvm on triangular meshes for elliptic equations we proved the 3rd order superconvergence rate of the gradient approximation u h u i 1 o h 3 and the 4th order superconvergence rate of the function value approximation u h u i 0 o h 4 for the quadratic fvm on triangular meshes here u h is the fvm solution and u i is the piecewise quadratic lagrange interpolation of the exact solution it should be pointed out that the superconvergence phenomena of fvm strongly depends on the construction of dual mesh specially for quadratic fvms the scheme presented in this paper is the unique scheme which holds superconvergence

unified analysis of finite volume methods for second order elliptic problems we establish a general framework for analyzing the class of finite volume methods which employ continuous or totally discontinuous trial functions and piecewise constant test functions under the framework optimal order convergence in the h 1 and l 2 norms can be obtained in a natural and systematic way for classical finite volume methods and new finite volume methods such as discontinuous finite volume methods applied to second order elliptic problems

a combined finite volume nonconforming mixed hybrid finite element scheme for degenerate parabolic problems we propose and analyze a numerical scheme for nonlinear degenerate parabolic convection diffusion reaction equations in two or three space dimensions we discretize the diffusion term which generally involves an inhomogeneous and anisotropic diffusion tensor over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming crouzeix raviart finite element method or using the stiffness matrix of the hybridization of the lowest order raviart thomas mixed finite element method the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh checking the local peclet number we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection dominated case this technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor we prove the convergence of the scheme only supposing the shape regularity condition for the original mesh we use a priori estimates and the kolmogorov relative compactness theorem for this purpose the proposed scheme is robust only 5 point 7 point in space dimension three locally conservative efficient and stable which is confirmed by numerical experiments

the semidiscrete finite volume element method for nonlinear convection diffusion problem abstract in this work a semidiscrete finite volume element method for the nonlinear convection diffusion problem is considered under some regular assumptions we obtain the l2 and h1 norm error estimates of numerical solution furthermore we investigate two grid finite volume element method for the considered equations compared with the standard method the two grid method is of the same order as the standard method in the h1 norm as long as the mesh sizes satisfy h o h 1 3 however the two grid method involves much less work than the standard method finally some numerical results are provided to verify the established theoretical analysis

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 q1 finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh abstract in this paper we study a q 1 finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh it is known that the coercivity is the basement for some other theoretical results stability h 1 and l 2 error estimates etc and the existing results were mainly obtained on h 1 parallelogram meshes with scalar diffusion coefficients for the cases of full diffusion tensors and arbitrary convex quadrilateral meshes we obtain a necessary and sufficient condition for the positive definiteness of the cell matrix related to the cell bilinear form based on this result a sufficient condition is suggested to guarantee the coercivity of the scheme more interesting is that this sufficient condition covers the traditional h 1 parallelogram mesh assumption and has an explicit expression by which one can easily judge on any diffusion tensor and any mesh with arbitrary mesh size h 0 moreover an h 1 error estimate is obtained without the h 1 parallelogram assumption and some numerical results are also provided to validate the theoretical results

piecewise linear petrov galerkin error estimates for the box method the perpendicular bisector box method pbm discretization of the laplacian on a simplicial element is equivalent to a piecewise linear petrov galerkin method with piecewise constant test functions this pbm element matrix is expressed in terms of galerkin s piecewise linear plg element matrix and the average values assumed by test functions over element faces only these results are shown to be applicable to the accuracy analysis of pbm discretizations of extremum stable elliptic gradient equations subject to regularity conditions on variable coefficients in these equations these gradient equations are shown to satisfy sufficient conditions to validate a discrete extremum principle on n dimensional delaunay triangulations this conclusion need not hold for the plg equations the dependence of the discretized laplacian on only the element face averages of test functions enables the substitution of piecewise polynomial test functions for the piecewise constant functions the latter are used to establish piecewise linear accuracy for the pbm by an adaptation of an error estimate for petrov galerkin methods by strang

linear independence of the t spline blending functions associated with some particular t meshes based on the local refinement algorithm addressed in sederberg et al 2004 17 we analyze the linear independence of the bi cubic t spline blending functions corresponding to particular t meshes

a finite difference scheme for two dimensional semiconductor device of heat conduction on composite triangular grids abstract the momentary state of a semiconductor device of heat conduction is described by a system of four nonlinear partial differential equations one elliptic equation is for the electrostatic two parabolic equations are for the electron concentration and the hole concentration and one heat exchange equation is for the temperature according to the necessary of practical numerical simulations and based on the balance equation finite difference schemes for two dimensional transient behavior of a semiconductor device of heat conduction on composite triangular grids are constructed studying their stability and convergence properties the error estimate in the energy norm is obtained finally a numerical example is given

convergence of finite volume schemes for poisson s equation on nonuniform meshes an error analysis of a family of cell center finite volume schemes for poisson s equation on cartesian product nonuniform meshes in two dimensions is presented optimal order error estimates are derived in the discrete h 1 norm under minimum smoothness requirements on the exact solution and without any additional assumption on the regularity of the mesh on quasi uniform meshes analogous estimates are obtained in the maximum norm

preconditioning the bidomain model with almost linear complexity the bidomain model is widely used in electro cardiology to simulate spreading of excitation in the myocardium and electrocardiograms it consists of a system of two parabolic reaction diffusion equations coupled with an ode system its discretisation displays an ill conditioned system matrix to be inverted at each time step simulations based on the bidomain model therefore are associated with high computational costs in this paper we propose a preconditioning for the bidomain model either for an isolated heart or in an extended framework including a coupling with the surrounding tissues the torso the preconditioning is based on a formulation of the discrete problem that is shown to be symmetric positive semi definite a block lu decomposition of the system together with a heuristic approximation referred to as the monodomain approximation are the key ingredients for the preconditioning definition numerical results are provided for two test cases a 2d test case on a realistic slice of the thorax based on a segmented heart medical image geometry a 3d test case involving a small cubic slab of tissue with orthotropic anisotropy the analysis of the resulting computational cost both in terms of cpu time and of iteration number shows an almost linear complexity with the problem size i e of type nlog a n for some constant a which is optimal complexity for such problems

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

a new class of high order finite volume methods for second order elliptic equations in the numerical simulation of many practical problems in physics and engineering finite volume methods are an important and popular class of discretization methods due to the local conservation and the capability of discretizing domains with complex geometry however they are limited by low order approximation since most existing finite volume methods use piecewise constant or linear function space to approximate the solution in this paper a new class of high order finite volume methods for second order elliptic equations is developed by combining high order finite element methods and linear finite volume methods optimal convergence rate in h 1 norm for our new quadratic finite volume methods over two dimensional triangular or rectangular grids is obtained

averaging and deterministic optimal control averaging is often used in ordinary differential equations when dealing with fast periodic phenomena it is shown here that it can be used efficiently in optimal control as the period tends to zero a limit or averaged problem is defined the open loop optimal control of the limit problem induces a cost which is optimal up to the second order when evaluated through the original dynamics the definition of the averaged problem is then generalized to the nonperiodic case it is shown that the bellman function of the original fast problem tends uniformly on any compact set to that of the averaged problem

penalty combinations of the ritz galerkin and finite difference methods for singularity problems abstract penalty combination of the ritz galerkin and finite difference methods is presented for solving elliptic boundary value problems with singularities the superconvergence rate o h2 of solution derivatives by the combination can be achieved while using quasiuniform rectangular difference grids where h is the maximal mesh length of difference grids used in the finite difference method and 0 is an arbitrarily small number it is due to its simplicity that the penalty combination of the ritz galerkin and finite difference methods is highly recommended for solving the complicated problems with multiple singularities and multiple interfaces

superconvergence of finite volume methods for the second order elliptic problem abstract this paper derives a superconvergence result for finite volume approximations of the second order elliptic problem by using a l2 projection post processing technique the superconvergence result is applicable to different kind of finite volume methods and to general quasi uniform meshes

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 coupling of finite volume and discontinuous galerkin method for elliptic problems the coupling of cell centered finite volume method with primal discontinuous galerkin method is introduced in this paper for elliptic problems convergence of the method with respect to the mesh size is proved numerical examples confirm the theoretical rates of convergence advantages of the coupled scheme are shown for problems with discontinuous coefficients or anisotropic diffusion matrix

high order characteristic finite volume methods for aerosol dynamic equations abstract aerosol particles have an important effect on changing of climate and human health where aerosols scatter and absorb the incoming solar radiation and thus decrease the precipitation efficiency of warm clouds and can cause an indirect radiative forcing associated with changes in cloud properties meanwhile it has also been recognized that the particles of aerosols in the sub micrometer size range can be inhaled and thus pose certain health hazards in this paper we analyze the finite volume method based on linear interpolation and hermite interpolation combined with the method of characteristics for the nonlinear aerosol dynamic equations on time and particle size which involve the advection process condensation process and the nonlinear coagulation process numerical experiments for the multiple log normal aerosol distributions are further given to confirm the theoretical results

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

block iterative solvers for higher order finite volume methods recently new higher order finite volume methods fvm were introduced in z cai j douglas m park development and analysis of higher order finite volume methods over rectangles for elliptic equations adv comput math 19 2003 3 33 where the linear system derived by the hybridization with lagrange multiplier satisfying the flux consistency condition is reduced to a linear system for a pressure variable by an appropriate quadrature rule we study the convergence of an iterative solver for this linear system the conjugate gradient cg method is a natural choice to solve the system but it seems slow possibly due to the non diagonal dominance of the system in this paper we propose block iterative methods with a reordering scheme to solve the linear system derived by the higher order fvm and prove their convergence with a proper ordering each block subproblem can be solved by fast methods such as the multigrid mg method the numerical experiments show that these block iterative methods are much faster than cg

two low order characteristic finite element methods for a convection dominated transport problem two low order rectangular finite elements for the convection diffusion problem with a modified characteristic finite element scheme are studied in this paper the o h 2 order error estimates in l 2 norm with respect to the space are obtained for one element with regular meshes and the other under anisotropic meshes in the process we use some distinct properties of the interpolation operator the integral identity technique and the mean value technique instead of the traditional elliptic projection which is an indispensable tool in the convergence analysis of the previous literature finally some numerical results are provided to verify our theoretical analysis

error estimates in l 2 h 1 and l in covolume methods for elliptic and parabolic p roblems a unified approach in this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane we introduce a general approach for connecting these methods with finite element method analysis this unified approach is used to prove known convergence results in the h 1 l 2 norms and new results in the max norm for the elliptic problems we demonstrate that the error u u h between the exact solution u and the approximate solution u h in the maximum norm is o h 2 ln h in the linear element case furthermore the maximur norm error in the gradient is shown to be of first order similar results hold for the parabolic problems

on the accuracy of the finite volume element method based on piecewise linear polynomials we present a general error estimation framework for a finite volume element fve method based on linear polynomials for solving second order elliptic boundary value problems this framework treats the fve method as a perturbation of the galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of fve methods in particular the error estimates and counterexamples in this paper will confirm that the fve method cannot have the standard o h2 convergence rate in the l2 norm when the source term has the minimum regularity only being in l2 even if the exact solution is in h2

on the heterogeneous multiscale method with various macroscopic solvers abstract the heterogeneous multiscale method hmm is a general method for efficient numerical solution of problems with multiscales it consists of two components an overall macroscopic solver for macrovariables on a macrogrid and an estimation of the missing macroscopic data from the microscopic model in this paper we present a state of the art review of the hmm with various macroscopic solvers including finite differences finite elements discontinuous galerkin mixed finite elements control volume finite elements nonconforming finite elements and mixed covolumes the first four solvers have been studied in the hmm setting the others are not as example the hmm with the nonconforming finite element macroscopic solver for nonlinear and random homogenization problems is also studied here

2 d and 3 d interpretation of electrical tomography measurements part 2 the inverse problem the interpretation of borehole to borehole electrical measurements requires solving an inverse problem for a given class of model geometries the usual approach to an inverse problem includes a model dependent task i e forward modeling and a generic task i e an optimization scheme we have developed an optimization algorithm using a nonlinear inversion technique this algorithm allows recovery of a possible resistivity distribution in an investigated zone between two boreholes or in the vicinity of them this resistivity distribution is defined as a set of 2 d or 3 d volumes of constant resistivity the inversion procedure minimizes a least squares term plus a damping term this latter term seeks to minimize the roughness of the solution an improved form of this smoothness term may enhance the spatial resolution of the resistivity image assuming that the resistivity contrast is known a priori this reconstruction algorithm has been tested for both 2 d and 3 d geometries these inversion tests we

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

simultaneous multiresolution strategies for nonrigid image registration multiresolution strategies are commonly used in the nonrigid registration to avoid local minima in the optimization space generally a step by step hierarchical approach is adopted in which the registration starts on a level with reduced complexity downsampled images global transformations then continuing to levels with increased complexity until the finest level is reached in this paper we propose two alternative multiresolution strategies for both the data and transformation models in which different resolution levels are considered simultaneously instead of subsequently through combining the different strategies for data and transformation we systematically define 3 3 multiresolution schemes including both existing and novel methods experiments on 10 pairs of computed tomography lung data sets showed that the best performing strategy resulted in a reduction of the upper quartile of the mean target registration error from 2 to 1 5 mm compared with the conventionally hierarchical multiresolution method while achieving smoother deformations experiments with intersubject registration of 18 3d t1 weighted mri brain scans confirmed that simultaneous multiresolution strategies produce more accurate registration results median of mean overlap increased from 0 55 to 0 57 and smoother deformation fields than the traditionally hierarchical method evaluation of robustness indicated that the largest differences in accuracy between methods are observed for structures with a relatively large initial misalignment

symmetric finite volume discretizations for parabolic problems based on a linear finite element space in this paper one semi discrete symmetric finite volume scheme and two full discrete symmetric finite volume schemes for a class of parabolic problems are presented and analyzed some numerical examples are reported

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

a symmetric characteristic fve method with second order accuracy for nonlinear convection diffusion problems the finite volume element fve methods used currently are essentially low order and unsymmetric in this paper by biquadratic elements and multistep methods we construct a second order fve scheme for nonlinear convection diffusion problem on nonuniform rectangular meshes to overcome the numerical oscillation we discretize the problem along its characteristic direction the choice of alternating direction strategy is critical in this paper which guarantees the high efficiency and symmetry of the discrete scheme optimal order error estimates in h 1 norm are derived and a numerical example is given at the end to confirm the usefulness of the method

superconvergence of coupling techniques in combined methods for elliptic equations with singularities abstract several coupling techniques such as the nonconforming constraints penalty and hybrid integrals of the ritz galerkin and finite difference methods are presented for solving elliptic boundary value problems with singularities based on suitable norms involving discrete solutions at specific points superconvergence rates on solution derivatives are exploited by using five combinations e g the nonconforming combination the penalty combination combinations i and ii and symmetric combination for quasi uniform rectangular grids the superconvergence rates o h 2 of solution derivatives by all five combinations can be achieved where h is the maximal mesh length of difference grids used in the finite difference method and 0 is an arbitrarily small number superconvergence analysis in this paper lies in estimates on error bounds caused by the coupling techniques and their incorporation with finite difference methods therefore a similar analysis and conclusions may be extended to linear finite element methods using triangulation by referring to existing references moreover the five combinations having o h 2 of solution derivatives are well suited to solving engineering problems with multiple singularities and multiple interfaces

the characteristic finite volume element method for the nonlinear convection dominated diffusion problem in modern numerical simulation of prospecting and exploiting oil gas resources and environmental science it is important to consider a numerical method for nonlinear convection dominated diffusion problems based on actual conditions such as the three dimensional characteristics of large scale science engineering computation we present a kind of characteristic finite volume element method some techniques such as calculus of variations commutating operators the theory of prior estimates and techniques are adopted suboptimal order error estimate in l 2 norm and optimal order error estimate in h 1 norm are derived to determine the errors for the approximate solution numerical results are presented to verify the performance of the scheme

the upwind finite volume element method based on straight triangular prism partition for nonlinear convection diffusion problem the upwind finite volume element method based on straight triangular prism partition is put forward for the nonlinear convection diffusion problems some techniques such as calculus of variations commutating operator and prior estimates are adopted optimal order error estimate in l 2 is derived to determine the error in approximate solution numerical experiment shows that the method is effective for avoiding numerical diffusion and nonphysical oscillations and reducing work time

on the construction and analysis of high order locally conservative finite volume type methods for one dimensional elliptic problems locally conservative finite volume type methods based on continuous piecewise polynomial functions of degree r ge 2 are introduced and analyzed in the context of indefinite elliptic problems in one space dimension the new methods extend and generalize the classical finite volume method based on piecewise linear functions we derive a priori error estimates in the l2 h1 and l infty norm and discuss superconvergence effects for the error and its derivative explicit residual based a posteriori error bounds in the l2 and energy norm are also derived we compute the experimental order of convergence and show the results of an adaptive algorithm based on the a posteriori error estimates

mixed covolume methods for quasi linear second order elliptic problems we consider covolume methods for the mixed formulations of quasi linear second order elliptic problems covolume methods for the mixed formulations of linear elliptic problem was first considered by russell rigorous block centered discretizations on irregular grids improved simulation of complex reservoir systems tech report 3 project report reservoir simulation research corporation tulsa ok 1995 and tested extensively in cai et al comput geosci 1 1997 pp 289 315 jones a mixed finite volume element method for accurate computation of fluid velocities in porous media ph d thesis university of colorado denver 1995 the analysis was carried out by chou and kwak siam j numer anal 37 2000 pp 758 771 for linear symmetric problems where they showed optimal error estimates in l2 norm for the pressure and in h div norm for the velocity in this paper we extend their results to quasi linear problems by following milner s argument math comp 44 1985 pp 303 320 through an adaptation of the duality argument of douglas and roberts math comp 44 1985 pp 39 52 for mixed covolume methods