fully conservative finite difference scheme in cylindrical coordinates for incompressible flow simulations a new finite difference scheme on a non uniform staggered grid in cylindrical coordinates is proposed for incompressible flow the scheme conserves both momentum and kinetic energy for inviscid flow with the exception of the time marching error provided that the discrete continuity equation is satisfied a novel pole treatment is also introduced where a discrete radial momentum equation with the fully conservative convection scheme is introduced at the pole the pole singularity is removed properly using analytical and numerical techniques the kinetic energy conservation property is tested for the inviscid concentric annular flow for the proposed and existing staggered finite difference schemes in cylindrical coordinates the pole treatment is verified for inviscid pipe flow mixed second and high order finite difference scheme is also proposed and the effect of the order of accuracy is demonstrated for the large eddy simulation of turbulent pipe flow

a localized re initialization equation for the conservative level set method the conservative level set methodology for interface transport is modified to allow for localized level set re initialization this approach is suitable to applications in which there is a significant amount of spatial variability in level set transport the steady state solution of the modified re initialization equation matches that of the original conservative level set provided an additional eikonal equation is solved which can be done efficiently through a fast marching method fmm implemented within the context of the accurate conservative level set method acls desjardins et al 2008 6 the fmm solution of this eikonal equation comes at no additional cost a metric for the appropriate amount of local re initialization is proposed based on estimates of local flow deformation and numerical diffusion the method is compared to standard global re initialization for two test cases yielding the expected results that minor differences are observed for zalesak s disk and improvements in both mass conservation and interface topology are seen for a drop deforming in a vortex finally the method is applied to simulation of a viscously damped standing wave and a three dimensional drop impacting on a shallow pool negligible differences are observed for the standing wave as expected for the last case results suggest that spatially varying re initialization provides a reduction in spurious interfacial corrugations improvements in the prediction of radial growth of the splashing lamella and a reduction in conservation errors as well as a reduction in overall computational cost that comes from improved conditioning of the pressure poisson equation due to the removal of spurious corrugations

a fast poisson solver for complex geometries robust fast solvers for the poisson equation have generally been limited to regular geometries where direct methods based on fourier analysis or cyclic reduction and multigrid methods can be used while multigrid methods can be applied in irregular domains and to a broader class of partial differential equations they are difficult to implement in a robust fashion since they require an appropriate hierarchy of coarse grids which are not provided in many practical situations in this paper we present a new fast poisson solver based on potential theory rather than on direct discretization of the partial differential equation our method combines fast algorithms for computing volume integrals and evaluating layer potentials on a grid with a fast multipole accelerated integral equation solver the amount of work required is o m log m n where m is the number of interior grid points and n is the number of points on the boundary asymptotically the cost of our method is just twice that of a standard poisson solver on a rectangular domain in which the problem domain can be embedded independent of the complexity of the geometry

computing interface motion in compressible gas dynamics a hamilton jacobi level set formulation of the equations of motion for propagating interfaces has been introduced recently by osher and sethian this formulation allows fronts to self intersect develop singularities and change topology the numerical algorithms based on this approach handle topological merging and breaking naturally work in any number of space dimensions and do not require that the moving front be written as a function instead the moving front is embedded as a particular level set in a fixed domain partial differential equation numerical techniques borrowed from hyperbolic conservation laws are then used to accurately capture the complicated surface motion that satisfies the global entropy condition for propagating fronts given by sethian in this paper we analyze the coupling of this level set formulation to a system of conservation laws for compressible gas dynamics we study both conservative and non conservative differencing of the level set function and compare the two approaches to illustrate the capability of the method we study the compressible rayleigh taylor and kelvin helmholtz instabilities for air air and air helium boundaries we perform numerical convergence studies of the method over a range of parameters and analyze the accuracy of this approach applied to these problems

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

a fast marching approach to multidimensional extrapolation a computationally efficient approach to extrapolating a data field with second order accuracy is presented this is achieved through the sequential solution of non homogeneous linear static hamilton jacobi equations which can be performed rapidly using the fast marching methodology in particular the method relies on a fast marching calculation of the distance from the manifold c that separates the subdomain w i n over which the quanity is known from the subdomain w o u t over which the quantity is to be extrapolated a parallel algorithm is included and discussed in the appendices results are compared to the multidimensional partial differential equation pde extrapolation approach of aslam aslam 2004 31 it is shown that the rate of convergence of the extrapolation within a narrow band near c is controlled by both the number of successive extrapolations performed and the order of accuracy of the spatial discretization for m successive extrapolating steps and a spatial discretization scheme of order n the rate of convergence in a narrow band is shown to be min n 1 m 1 results show that for a wide range of error levels the fast marching extrapolation strategy leads to dramatic improvements in computational cost when compared to the pde approach

on the multiplicity of the solution set of some nonlinear boundary value problems ii on considere le probleme non lineaire lu f u s 1 h dans u 0 sur soit h l 2 f est de classe c et que f 1 et n f n 1 n 1 alors il existe s 0 tel que pour tout s s 0 l equation a au moins 2 solutions si n est impair et 3 solutions si n est pair

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

a conservative level set method for two phase flow ii in this paper we continue to develop and study the conservative level set method for incompressible two phase flow with surface tension introduced in j comput phys 210 2005 225 246 we formulate a modification of the reinitialization and present a theoretical study of what kind of conservation we can expect of the method a finite element discretization is presented as well as an adaptive mesh control procedure numerical experiments relevant for problems in petroleum engineering and material science are presented for these problems the surface tension is strong and conservation of mass is important problems in both two and three dimensions with uniform as well as non uniform grids are studied from these calculations convergence and conservation is studied good conservation and convergence are observed

semi implicit surface tension formulation with a lagrangian surface mesh on an eulerian simulation grid we present a method for applying semi implicit forces on a lagrangian mesh to an eulerian discretization of the navier stokes equations in a way that produces a sparse symmetric positive definite system the resulting method has semi implicit and fully coupled viscosity pressure and lagrangian forces we apply our new framework for forces on a lagrangian mesh to the case of a surface tension force which when treated explicitly leads to a tight time step restriction by applying surface tension as a semi implicit lagrangian force the resulting method benefits from improved stability and the ability to take larger time steps the resulting discretization is also able to maintain parasitic currents at low levels

alternating direction ghost fluid methods for solving the heat equation with interfaces abstract this work presents two new alternating direction implicit adi schemes for solving parabolic interface problems in two and three dimensions first the ghost fluid method gfm is adopted for the first time in the literature to treat interface jumps in the adi framework which results in symmetric and tridiagonal finite difference matrices in each adi step the proposed gfm adi scheme achieves a first order of accuracy in both space and time as confirmed by numerical experiments involving complex interface shapes and spatial temporal dependent jumps the gfm adi scheme also maintains the adi efficiency the computational complexity for each time step scales as o n for a total degree of freedom n in higher dimensions second a new matched interface and boundary mib scheme is constructed which downgrades the quadratic polynomial bases in the existing second order mib to linear ones interestingly the resulting mib adi or madi scheme produces the same finite difference matrices as the gfm adi scheme in all dimensions hence the madi scheme can be regarded as an improvement of the gfm because it calculates tangential jumps which are omitted in the gfm consequently the present madi scheme is constantly more accurate than the gfm adi in all numerical examples while keeping the same computational efficiency nevertheless the madi scheme is semi implicit due to tangential jump approximations while the gfm adi scheme is fully implicit without tangential corrections thus the gfm adi scheme could be more stable than the madi scheme when a huge contrast is presented in diffusion coefficients

multicomponent flow calculations by a consistent primitive algorithm the dynamics of inviscid multicomponent fluids may be modelled by the euler equations augmented by one or more additional species equation s attempts to compute solutions for extended euler models in conservation form show strong oscillations and other computational inaccuracies near material interfaces these are due to erroneous pressure fluctuations generated by the conservative wave model this problem does not occur in single component computations and arises only in the presence of several species a nonconservative primitive euler formulation is proposed which results in complete elimination of the oscillations the numerical algorithm uses small viscous perturbations to remove leading order conservation errors and is conservative to the order of numerical approximation numerical experiments show clean monotonic solution profiles with acceptably small conservation error for shocks of weak to moderate strengths

fully conservative higher order finite difference schemes for incompressible flow conservation properties of the mass momentum and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out in particular it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass momentum and kinetic energy this deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems finite difference schemes in a collocated grid system are also analyzed and a violation of kinetic energy conservation is revealed the predicted conservation properties are demonstrated numerically in simulations of inviscid white noise performed in a two dimensional periodic domain the proposed fourth order schemes in a staggered grid system are generalized for the case of a non uniform mesh and the resulting scheme is used to perform large eddy simulations of turbulent channel flow

on nonlinear oscillations in a suspension bridge system in this paper we study nonlinear oscillations in a suspension bridge system governed by two coupled nonlinear partial differential equations by applying the leray schauder degree theory it is proved that the suspension bridge system has at least two solutions one is a near equilibrium oscillation and the other is a large amplitude oscillation

a sharp interface method for incompressible two phase flows we present a sharp interface method for computing incompressible immiscible two phase flows it couples the level set and volume of fluid techniques and retains their advantages while overcoming their weaknesses it is stable and robust even for large density and viscosity ratios on the order of 1000 to 1 the numerical method is an extension of the second order method presented by sussman m sussman a second order coupled levelset and volume of fluid method for computing growth and collapse of vapor bubbles journal of computational physics 187 2003 110 136 in which the previous method treated the gas pressure as spatially constant and the present method treats the gas as a second incompressible fluid the new method yields solutions in the zero gas density limit which are comparable in accuracy to the method in which the gas pressure was treated as spatially constant this improvement in accuracy allows one to compute accurate solutions on relatively coarse grids thereby providing a speed up over continuum or ghost fluid methods

a hybrid immersed boundary and immersed interface method for electrohydrodynamic simulations in this paper we develop a hybrid immersed boundary ib and immersed interface method iim to simulate the dynamics of a drop under an electric field in navier stokes flows within the leaky dielectric framework with piecewise constant electric properties in each fluid the electric stress can be treated as an interfacial force on the drop interface thus both the electric and capillary forces can be formulated in a unified immersed boundary framework the electric potential satisfies a laplace equation which is solved numerically by an augmented immersed interface method which incorporates the jump conditions naturally along the normal direction the incompressible navier stokes equations for the fluids are solved using a projection method on a staggered mac grid and the potential is solved at the cell center the interface is tracked in a lagrangian manner with mesh control by adding an artificial tangential velocity to transport the lagrangian markers to ensure that the spacing between markers is uniform throughout the computations a series of numerical tests for the present scheme have been conducted to illustrate the accuracy and applicability of the method we first compute the potential and its gradient electric field to perform the accuracy check for the present augmented iim we then check the convergence of the interfacial electric force and the fluid variables we further run a series of simulations with different permittivity and conductivity ratios and compare with the results obtained by the small deformation theory and other numerical results in literature in addition we also study the electric effect for a drop under shear flow

mathematical analysis of dynamic models of suspension bridges in this paper we present some mathematical analysis of dynamic pde models of suspension bridges as proposed by lazer and mckenna our results are illustrated by numerical simulation with physical interpretation

two phase electrohydrodynamic simulations using a volume of fluid approach a numerical methodology to simulate two phase electrohydrodynamic flows under the volume of fluid paradigm is proposed the electric force in such systems acts only at the interface and is zero elsewhere in the two fluids continuum surface force representations are derived for the electric field force in a system of dielectric dielectric and conducting conducting fluids on the basis of analytical calculations for simple flow problems we propose a weighted harmonic mean interpolation scheme to smoothen the electric properties in the diffused transition region interface it is shown that a wrong choice of interpolation scheme weighted arithmetic mean may lead to a transition region thickness dependent electric field in the bulk we simulate a set of problems with exact or approximate analytical solutions to validate the numerical model proposed a coupled level set and volume of fluid clsvof algorithm has been used for simulations presented here

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

a new c1 finite element full heptic a new c1 finite element complete in 7th degree polynomial basis is described and compared with other c1 elements in an eigen elements analysis

c 1 curved finite elements and applications to plate and shell problems equation 1 curved finite elements and applications to plate and shell problems abstract many thin plate and thin shell problems are set on plane reference domains with a curved boundary their approximation by conforming finite elements methods requires c 1 curved finite elements entirely compatible with the associated c 1 rectilinear finite elements in this contribution we introduce a c 1 curved finite element compatible with the p5 argyris element we study its approximation properties and then we use such an element to approximate the solution of thin plate or thin shell problems set on a plane curved boundary domain we prove the convergence and we get a priori asymptotic error estimates which show the very high degree of accuracy of the method moreover we obtain criteria to observe when choosing the numerical integration schemes in order to preserve the order of the error estimates obtained for exact integration

a note on immersed interface method for three dimensional elliptic equations abstract the immersed interface method proposed by leveque and li 1 is extended to three dimensional elliptic equations of the form x u x x u x f x we study the situation in which there is an irregular interface surface s contained in the solution domain across which and f may be discontinuous or even singular as a result the solution u will usually be nonsmooth or even discontinuous a finite difference approach with a uniform cartesian grid is used in the discretization local truncation error analysis is performed to estimate the accuracy of the numerical solution

connectivity free front tracking method for multiphase flows with free surfaces in this study a connectivity free front tracking method is developed to simulate multiphase flows with free surfaces this method is based on the point set method which does not require any connectivities between interfacial points to represent the interface the main advantage of the connectivity free approach is the easiness in re constructing the interface when large topology change occurs it requires an indicator field to be constructed first based on the existing interface and the surface curvature and normal are then computed using the indicator field here we adopt the reproducing kernel particle method rkpm interpolation function that provides the ability to deal with free surface flows and the flexibility of using non uniform meshes when local fine resolution is needed a points regeneration scheme is developed to construct smooth interfaces and to automatically handle topology changes the mass conservation is verified by performing a single vortex advection test several 2 d and 3 d numerical tests including an oscillating droplet dam breaking two droplet impacting and multi bubble merging are presented to show the accuracy and the robustness of the method

algebraic multigrid solvers for complex valued matrices in the mathematical modeling of real life applications systems of equations with complex coefficients often arise while many techniques of numerical linear algebra e g krylov subspace methods extend directly to the case of complex valued matrices some of the most effective preconditioning techniques and linear solvers are limited to the real valued case here we consider the extension of the popular algebraic multigrid method to such complex valued systems the choices for this generalization are motivated by classical multigrid considerations evaluated with the tools of local fourier analysis and verified on a selection of problems related to real life applications

rapid parallel evaluation of integrals in potential theory on general three dimensional regions we present a new high order accurate method for the rapid parallel evaluation of certain integrals in potential theory on general three dimensional regions these methods use fast methods for solving the differential equation which the kernel satisfies and the number of operations needed to evaluate the integrals is essentially equal to the number of operations needed to solve the differential equation on a regular rectangular grid in particular one can evaluate integrals whose kernels are the greens function for poissons equation by using fourier methods on a rectangular grid or a fast poisson solver thus these methods avoid the problems associated with using quadrature methods to evaluate an integral with a singular kernel numerical results are presented for experiments on a variety of geometries

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

curved finite elements of class implementation and numerical experiments part 1 construction and numerical tests of the interpolation properties the aim of this work is to show how to implement some of the methods of curved finite elements of class c1 introduced and analyzed in a previous work by the first author these methods use curved finite elements which are c1 compatible with the argyris triangle a careful description of the implementation is completed by some numerical experiments which show the very high degree of accuracy of the associated interpolation subsequently in a second part we will show the efficiency of such methods for solving thin plate or thin shell problems set on curved boundary domains

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

high order finite difference schemes on non uniform meshes with good conservation properties numerical simulation of turbulent flows dns or les requires numerical methods that are both stable and free of numerical dissipation one way to achieve this is to enforce additional constraints such as discrete conservation of mass momentum and kinetic energy the objective of this work is to generalize the high order schemes of morinishi et al to non uniform meshes while maintaining conservation properties of the schemes as much as possible this generalization is achieved by preserving symmetries of the uniform mesh case the proposed schemes do not simultaneously conserve mass momentum and kinetic energy however depending on the form of the convective term conservation of either momentum or energy in addition to mass can be achieved it is shown that the conservation properties of the generalized schemes are as good as those of the standard second order finite difference scheme on non uniform meshes while the accuracy of the new schemes is definitely superior the predicted conservation properties are demonstrated numerically in inviscid flow simulations

β1 curved finite elements with numerical integration for thin plate and thin shell problems part 2 approximation of thin plate and thin shell problems abstract in this paper we study the approximation of solutions of linear thin shell problems formulated on plane reference domains with curved boundary according to the model of w t koiter sufficient conditions are given which preserve the order of convergence of the method these conditions involve both the approximation of the components of the displacement with straight and curved finite element families of class 1 and the degree of accuracy of the numerical quadrature schemes we conclude by examining various examples as a particular case we apply the results to plate bending problems

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

multiple periodic solutions for suspension bridge systems abstract this paper studies the existence of multiple periodic oscillations of the suspension bridge system governed by the coupled nonlinear beam and wave equations by using the degree theory it is proved that the system has at least three periodic solutions

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow

a front tracking method for the computations of multiphase flow direct numerical simulations of multiphase flows using a front tracking method are presented the method is based on writing one set of governing equations for the whole computational domain and treating the different phases as one fluid with variable material properties interfacial terms are accounted for by adding the appropriate sources as functions at the boundary separating the phases the unsteady navier stokes equations are solved by a conventional finite volume method on a fixed structured grid and the interface or front is tracked explicitly by connected marker points interfacial source terms such as surface tension are computed on the front and transferred to the fixed grid advection of fluid properties such as density is done by following the motion of the front the method has been implemented for fully three dimensional flows as well as for two dimensional and axisymmetric ones first the method is described for the flow of two or more isothermal phases the representation of the moving interface and its dynamic restructuring as well as the transfer of information between the moving front and the fixed grid are discussed applications and extensions of the method to homogeneous bubbly flows atomization flows with variable surface tension solidification and boiling are then presented

improved volume conservation in the computation of flows with immersed elastic boundaries this paper introduces a recipe for the construction of a finite difference divergence operator the coefficients of which are completely determined once an interpolation scheme has been chosen substitution of this divergence operator for the previously used divergence based on central differences makes a dramatic improvement in the overall volume conservation that is observed in an immersed boundary computation this improvement is particularly important for computations in which an elastic boundary separates chambers containing fluid at substantially different pressures a situation which is prominent in cardiac fluid dynamics during the contraction of the ventricles

a ghost fluid level set methodology for simulating multiphase electrohydrodynamic flows with application to liquid fuel injection in this paper we present the development of a sharp numerical scheme for multiphase electrohydrodynamic ehd flows for a high electric reynolds number regime the electric potential poisson equation contains ehd interface boundary conditions which are implemented using the ghost fluid method gfm the gfm is also used to solve the pressure poisson equation the methods detailed here are integrated with state of the art interface transport techniques and coupled to a robust high order fully conservative finite difference navier stokes solver test cases with exact or approximate analytic solutions are used to assess the robustness and accuracy of the ehd numerical scheme the method is then applied to simulate a charged liquid kerosene jet

the rapid evaluation of volume integrals of potential theory on general regions abstract we present a new method for the rapid high order accurate evaluation of certain volume integrals in potential theory on general irregular regions the kernels of the integrals are either a fundamental solution or a linear combination of the derivatives of a fundamental solution of a second order linear elliptic differential equation instead of using a standard quadrature formula or the exact evaluation of any integral the methods rely on rapid methods of solving the differential equation which the kernel is the solution of therefore the number of operations needed to evaluate the volume integral is essentially equal to the number of operations needed to solve the differential equation on a rectangular region with a regular grid and the method requires no evaluation of the kernel