the chain collocation method a spectrally accurate calculus of forms preserving in the discrete realm the underlying geometric topological and algebraic structures at stake in partial differential equations has proven to be a fruitful guiding principle for numerical methods in a variety of fields such as elasticity electromagnetism or fluid mechanics however structure preserving methods have traditionally used spaces of piecewise polynomial basis functions for differential forms yet in many problems where solutions are smoothly varying in space a spectral numerical treatment is called for in an effort to provide structure preserving numerical tools with spectral accuracy on logically rectangular grids over periodic or bounded domains we present a spectral extension of the discrete exterior calculus dec with resulting computational tools extending well known collocation based spectral methods its efficient implementation using fast fourier transforms is provided as well

the numerical range theory and boundedness of solutions of the complementarity problem abstract using the numerical range theory of an operator defined on a hilbert space we study the boundedness of the solution set for the complementarity problem defined in a hilbert space

the orthogonal decomposition theorems for mimetic finite difference methods accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus provide reliable finite difference methods for approximating the solutions to a wide class of partial differential equations these methods mimic many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields i e they are divergence free and should satisfy a discrete version of the orthogonal decomposition theorem this theorem plays a fundamental role in the theory of generalized solutions and in the numerical solution of physical models including the navier stokes equations and in electrodynamics we are deriving mimetic finite difference approximations of the divergence gradient and curl that satisfy discrete analogs of the integral identities satisfied by the differential operators we first define the natural discrete divergence gradient and curl operators based on coordinate invariant definitions such as gauss s theorem for the divergence next we use the formal adjoints of these natural operators to derive compatible divergence gradient and curl operators with complementary domains and ranges of values in this paper we prove that these operators satisfy discrete analogs of the orthogonal decomposition theorem and demonstrate how a discrete vector can be decomposed into two orthogonal vectors in a unique way satisfying a discrete analog of the formula vec a ggrad varphi curl vec b we also present a numerical example to illustrate the numerical procedure and calculate the convergence rate of the method for a spiral vector field

convergence analysis of a finite volume method for maxwell s equations in nonhomogeneous media in this paper we analyze a recently developed finite volume method for the time dependent maxwell s equations in a three dimensional polyhedral domain composed of two dielectric materials with different parameter values for the electric permittivity and the magnetic permeability convergence and error estimates of the numerical scheme are established for general nonuniform tetrahedral triangulations of the physical domain in the case of nonuniform rectangular grids the scheme converges with second order accuracy in the discrete l2 norm despite the low regularity of the true solution over the entire domain in particular the finite volume method is shown to be superconvergent in the discrete h curl norm in addition the explicit dependence of the error estimates on the material parameters is given

the discrete duality finite volume method for stokes equations on three dimensional polyhedral meshes we develop a discrete duality finite volume method for the three dimensional steady stokes problem with a variable viscosity coefficient on polyhedral meshes under very general assumptions on the mesh which may admit nonconforming polyhedrons we prove the stability and well posedness of the scheme we also prove the convergence of the numerical approximation to the velocity velocity gradient and pressure and derive a priori estimates for the corresponding approximation error final numerical experiments confirm the theoretical predictions

a discontinuous galerkin least squares method for div curl systems abstract a discontinuous galerkin dg least squares finite element method is developed for solving a div curl problem this finite element method leads to a simple symmetric positive definite system and can work on a general polytopal mesh optimal convergence rate in the energy norm is established the numerical results confirm the proposed theory

a high order staggered meshless method for elliptic problems we present a new meshless method for scalar diffusion equations which is motivated by their compatible discretizations on primal dual grids unlike the latter though our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points em x i this graph defines a local primal dual grid complex with a virtual dual grid in the sense that specification of the dual metric attributes is implicit in the method s construction our method combines a topological gradient operator on the local primal grid with a generalized moving least squares approximation of the divergence on the local dual grid we show that the resulting approximation of the div grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method which attains o h m convergence in both l 2 and h 1 norms similar to mixed finite element methods we demonstrate this convergence on curvilinear domains using manufactured solutions in two and th

existence of solutions for a class of hemivariational inequality problems in this paper we are concerned with the existence of solutions for a class of hartman stampacchia type hemivariational inequalities by using the clarke generalized directional derivative and the galerkin approximation method two existence results of solutions for the generalized pseudomonotone mapping hemivariational inequality and elliptic hemivariational inequality are obtained

the mimetic finite difference method for the 3d magnetostatic field problems on polyhedral meshes we extend the mimetic finite difference mfd method to the numerical treatment of magnetostatic fields problems in mixed div curl form for the divergence free magnetic vector potential to accomplish this task we introduce three sets of degrees of freedom that are attached to the vertices the edges and the faces of the mesh and two discrete operators mimicking the curl and the gradient operator of the differential setting then we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the div curl variational formulation of the magnetostatic equations this construction is based on an algebraic consistency condition that generalizes the usual construction of the inner products of the mfd method we also discuss the linear algebraic form of the resulting mfd scheme its practical implementation and discuss existence and uniqueness of the numerical solution by generalizing the concept of logically rectangular or cubic meshes by hyman and shashkov to the case of unstructured polyhedral meshes the accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem

convergence analysis of the mimetic finite difference method for elliptic problems we propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes the a priori error analysis relies on the connection between the mimetic formulation and the lowest order raviart thomas mixed finite element method the theoretical results are confirmed by numerical experiments

analysis and computation of compatible least squares methods for div curl equations we develop and analyze least squares finite element methods for two complementary div curl elliptic boundary value problems the first one prescribes the tangential component of the vector field on the boundary and is solved using curl conforming elements the second problem specifies the normal component of the vector field and is handled by div conforming elements we prove that both least squares formulations are norm equivalent with respect to suitable discrete norms yield optimal asymptotic error estimates and give rise to algebraic systems that can be solved by efficient algebraic multigrid methods numerical results that illustrate scalability of iterative solvers and optimal rates of convergence are also included

discrete calculus methods for diffusion a general methodology for the solution of partial differential equations is described in which the discretization of the calculus is exact and all approximation occurs as an interpolation problem on the material constitutive equations the fact that the calculus is exact gives these methods the ability to capture the physics of pde systems well the construction of both node and cell based methods of first and second order are described for the problem of unsteady heat conduction though the method is applicable to any pde system the performance of these new methods are compared to classic solution methods on unstructured 2d and 3d meshes for a variety of simple and complex test cases

nonlinear complementarity problem and galerkin method etude du probleme de complementarite non lineaire dans des espaces de dimensions infinies en utilisant l approximation de galerkin et une generalisation de l hypothese de karamardian on obtient plusieurs resultats d existence pour une forme speciale du probleme ou f x x l x t x l etant un operateur lineaire et t un operateur non lineaire

a conservative treatment of zonal boundaries for euler equation calculations finite difference calculations require the generation of a grid for the region of interest a zonal approach wherein the given region is subdivided into zones and the grid for each zone is generated independently makes the grid generation process for complicated topologies and for regions requiring selective grid refinement a fairly simple task this approach results in new boundaries within the given region that is zonal boundaries at the interfaces of the various zones the zonal boundary scheme the integration scheme used to update the points on the zonal boundary for the euler equations must be conservative accurate stable and applicable to general curvilinear coordinate systems a zonal boundary scheme with these desirable properties is developed in this study the scheme is designed for explicit first order accurate integration schemes but can be modified to accommodate second order accurate explicit and implicit integration schemes results for inviscid flow including supersonic flow over a cylinder blast wave diffraction by a ramp and one dimensional shock tube flow are obtained on zonal grids the conservative nature of the zonal boundary scheme permits the smooth transition of the discontinuities associated with these flows from one zone to another the calculations also demonstrate the continuity of contour lines across zonal boundaries that can be achieved with the present zonal scheme

curvilinear immersed boundary method for simulating fluid structure interaction with complex 3d rigid bodies the sharp interface curvib approach of ge and sotiropoulos l ge f sotiropoulos a numerical method for solving the 3d unsteady incompressible navier stokes equations in curvilinear domains with complex immersed boundaries journal of computational physics 225 2007 1782 1809 is extended to simulate fluid structure interaction fsi problems involving complex 3d rigid bodies undergoing large structural displacements the fsi solver adopts the partitioned fsi solution approach and both loose and strong coupling strategies are implemented the interfaces between immersed bodies and the fluid are discretized with a lagrangian grid and tracked with an explicit front tracking approach an efficient ray tracing algorithm is developed to quickly identify the relationship between the background grid and the moving bodies numerical experiments are carried out for two fsi problems vortex induced vibration of elastically mounted cylinders and flow through a bileaflet mechanical heart valve at physiologic conditions for both cases the computed results are in excellent agreement with benchmark simulations and experimental measurements the numerical experiments suggest that both the properties of the structure mass geometry and the local flow conditions can play an important role in determining the stability of the fsi algorithm under certain conditions the fsi algorithm is unconditionally unstable even when strong coupling fsi is employed for such cases however combining the strong coupling iteration with under relaxation in conjunction with the aitken s acceleration technique is shown to effectively resolve the stability problems a theoretical analysis is presented to explain the findings of the numerical experiments it is shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the fsi algorithm the stabilizing role of under relaxation is also clarified and the upper bound of the under relaxation coefficient required for stability is derived

variational inequalities and fixed point theorems for pm maps abstract we study mapstfor whichj tis pseudo monotone we call suchta pm map this includes compact maps in suitable spaces and pseudo contractive maps in hilbert spaces we study variational inequalities in a situation which was previously done only whentis compact we show that our variational inequalities are well suited to treating existence of fixed points for generalized inward in particular weakly inward pm maps in hilbert spaces our new results on existence of fixed points generalizes many earlier results obtained by using other methods an application to homogeneous integral equations is provided

variational inequalities with generalized monotone operators we investigate the variational inequality with pseudomonotone operators in the sense of karamardian in banach spaces new existence results which extend many known results in infinite dimensional spaces are derived under rather weak assumptions new uniqueness results which also seem to be new even in finite dimensional spaces are also derived in particular new existence and uniqueness results for the complementarity problem with pseudomonotone operators in banach spaces are obtained also existence and uniqueness results for minimization problems with pseudoconvex functions in banach spaces are obtained

the discrete duality finite volume method for convection diffusion problems in this paper we extend the discrete duality finite volume ddfv formulation to the steady convection diffusion equation the discrete gradients defined in ddfv are used to define a cell based gradient for the control volumes of both the primal and dual meshes in order to achieve a higher order accurate numerical flux for the convection term a priori analysis is carried out to show convergence of the approximation and a global first order convergence rate is derived the theoretical results are confirmed by some numerical experiments

local mesh refinement within a multi block structured grid scheme for general flows abstract a multi level flow adaptive mesh refinement strategy has been formulated and implemented within a structured grid multiblock scheme for 2d laminar transitional and turbulent recirculating flow although the locally refined irregular grid portions are organised as block structured patches within coarse grid blocks a special data structure allows high storage economy to be achieved issues of particular interest in the study include the performance of different flow adaptation sensors based on solution properties and error and the response of the solution accuracy to inter grid interpolation practices based on bi linear bi quadratic and compact interpolation the effectiveness of the local refinement strategy especially in terms of cpu and memory resources is examined by reference to four test cases ranging from an analytically prescribed scalar field over a square domain to a transitional flow over a highly loaded compressor blade the study demonstrates that cpu and memory savings with a three level refinement for the compressor blade case are of order 70 and 90 respectively relative to the requirements of globally fine grids returning similar accuracy levels

a rbf partition of unity collocation method based on finite difference for initial boundary value problems abstract meshfree radial basis function rbf methods are popular tools used to numerically solve partial differential equations pdes they take advantage of being flexible with respect to geometry easy to implement in higher dimensions and can also provide high order convergence since one of the main disadvantages of global rbf based methods is generally the computational cost associated with the solution of large linear systems in this paper we focus on a localizing rbf partition of unity method rbf pum based on a finite difference fd scheme specifically we propose a new rbf pum fd collocation method which can successfully be applied to solve time dependent pdes this approach allows to significantly decrease ill conditioning of traditional rbf based methods moreover the rbf pum fd scheme results in a sparse matrix system reducing the computational effort but maintaining at the same time a high level of accuracy numerical experiments show performances of our collocation scheme on two benchmark problems involving unsteady convection diffusion and pseudo parabolic equations

exceptional family of elements and solvability of variational inequalities for mappings defined only on closed convex cones in banach spaces abstract in this paper we generalize the concept of exceptional family of elements for a completely continuous field from hilbert spaces to banach spaces and we study the solvability of the variational inequalities with respect to a mapping f that is from a closed convex cone of a banach space b to the dual space b by applying the generalized projection operator k and by using the leray schauder type alternative

a numerical method for large eddy simulation in complex geometries we discuss the development of a numerical algorithm and solver capable of performing large eddy simulation in the very complex geometries often encountered in industrial applications the algorithm is developed for unstructured hybrid grids is non dissipative yet robust at high reynolds numbers on highly skewed grids simulation results for a variety of flows are presented

discrete hodge operators many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms discrete differential forms provide a natural and canonical approach to their discretization however much freedom remains concerning the choice of discrete hodge operators that is discrete analogues of constitutive laws a generic discrete hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations we reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus

a fast lattice green s function method for solving viscous incompressible flows on unbounded domains a computationally efficient method for solving three dimensional viscous incompressible flows on unbounded domains is presented the method formally discretizes the incompressible navier stokes equations on an unbounded staggered cartesian grid operations are limited to a finite computational domain through a lattice green s function technique this technique obtains solutions to inhomogeneous difference equations through the discrete convolution of source terms with the fundamental solutions of the discrete operators the differential algebraic equations describing the temporal evolution of the discrete momentum equation and incompressibility constraint are numerically solved by combining an integrating factor technique for the viscous term and a half explicit runge kutta scheme for the convective term a projection method that exploits the mimetic and commutativity properties of the discrete operators is used to efficiently solve the system of equations that arises in each stage of the time integration scheme linear complexity fast computation rates and parallel scalability are achieved using recently developed fast multipole methods for difference equations the accuracy and physical fidelity of solutions are verified through numerical simulations of vortex rings

the discrete continuity equation in primitive variable solutions of incompressible flow abstract the use of a non staggered computational grid for the numerical solutions of the incompressible flow equations has many advantages over the use of a staggered grid a penalty however is inherent in the finite difference approximations of the governing equations on non staggered grids in the primitive variable solutions the penalty is that the discrete continuity equation does not converge to machine accuracy rather it converges to a source term which is proportional to the fourth order derivative of the pressure the time increment and the square of the grid spacing an approach which minimizes the error in the discrete continuity equation is developed numerical results obtained for the driven cavity problem confirm the analytical developments

a smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations the effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods the problem with this representation is that the volume force exhibits non physical oscillations in moving boundary simulations a smoothing technique for discrete delta functions has been developed in this paper to suppress the non physical oscillations in the volume forces we have found that the non physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions it has been shown that the smoothed discrete delta functions constructed in this paper have one order higher derivative than the regular ones moreover not only the smoothed discrete delta functions satisfy the first two discrete moment conditions but also their derivatives satisfy one order higher moment condition than the regular ones the smoothed discrete delta functions are tested by three test cases a one dimensional heat equation with a moving singular force a two dimensional flow past an oscillating cylinder and the vortex induced vibration of a cylinder the numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

a finite element method for first order elliptic systems in three dimensions abstract we consider finite element approximation schemes for boundary value problems for linear first order elliptic systems in three dimensions we also discuss div curl type systems in three independent variables both a least squares method and a new subdomain collocation least squares method are considered the analysis yield optimal asymptotic h 1 and l 2 error estimates

a discrete duality finite volume approach to hodge decomposition and div curl problems on almost arbitrary two dimensional meshes we define discrete differential operators such as gradient divergence and curl on general two dimensional nonorthogonal meshes these discrete operators verify discrete analogues of usual continuous theorems discrete green formulae discrete hodge decomposition of vector fields and vector curls have a vanishing divergence and gradients have a vanishing curl we apply these ideas to discretize div curl systems we give error estimates based on the reformulation of these systems into equivalent equations for the potentials numerical results illustrate the use of the method on several types of meshes some of which are degenerating triangular meshes and nonconforming locally refined meshes

a numerical method for solving the 3d unsteady incompressible navier stokes equations in curvilinear domains with complex immersed boundaries a novel numerical method is developed that integrates boundary conforming grids with a sharp interface immersed boundary methodology the method is intended for simulating internal flows containing complex moving immersed boundaries such as those encountered in several cardiovascular applications the background domain e g the empty aorta is discretized efficiently with a curvilinear boundary fitted mesh while the complex moving immersed boundary say a prosthetic heart valve is treated with the sharp interface hybrid cartesian immersed boundary approach of gilmanov and sotiropoulos a gilmanov f sotiropoulos a hybrid cartesian immersed boundary method for simulating flows with 3d geometrically complex moving bodies journal of computational physics 207 2005 457 492 to facilitate the implementation of this novel modeling paradigm in complex flow simulations an accurate and efficient numerical method is developed for solving the unsteady incompressible navier stokes equations in generalized curvilinear coordinates the method employs a novel fully curvilinear staggered grid discretization approach which does not require either the explicit evaluation of the christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations the equations are integrated in time using an efficient second order accurate fractional step methodology coupled with a jacobian free newton krylov solver for the momentum equations and a gmres solver enhanced with multigrid as preconditioner for the poisson equation several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady pulsatile flow through a curved pipe bend to demonstrate the ability of the method to simulate flows with complex moving immersed boundaries we apply it to calculate pulsatile physiological flow through a mechanical bileaflet heart valve mounted in a model straight aorta with an anatomical like triple sinus

construction of a finite element basis of the first de rham cohomology group and numerical solution of 3d magnetostatic problems we devise an efficient algorithm for the finite element approximation of harmonic fields and the numerical solution of three dimensional magnetostatic problems in particular we construct a finite element basis of the first de rham cohomology group of the computational domain the proposed method works for general topological configurations and does not need the determination of cutting surfaces

direct discretization of planar div curl problems a control volume method is proposed for planar div curl systems the method is independent of potential and least squares formulations and works directly with the div curl system the novelty of the technique lies in its use of a single local vector field component and two control volumes rather than the other way round a discrete vector field theory comes quite naturally from this idea and is developed in the paper error estimates are proved for the method and other ramifications investigated

multiblock hybrid grid finite volume method to solve flow in irregular geometries in this work a finite volume based finite element method is suitably developed for solving incompressible flow and heat transfer on collocated hybrid grid topologies the method is generally applicable to arbitrarily shaped elements and orientations and thus challenges the potential to unify many of the different grid topologies into a single formulation the key point in this formulation is the correct estimation of the convective and diffusive fluxes at the cell faces using a novel physical influence scheme this scheme remarkably enhances the achieved solution accuracy it is shown that the extended formulation is robust enough to treat any combination of multiblock meshes with dual element shape employments in this regard the solution domain is broken up into a number of different multiblock arrangements in which each block is filled with only one type of finite element shape the combined grid can decrease the computational time and memory requirements and increase the numerical accuracy the results of the extended formulation are validated against different benchmark and other solutions the current results are in excellent agreement with the other solutions without exhibiting any disturbances around the block boundaries

a high order scheme for unsteady heat conduction equations abstract in this work a high order scheme is proposed for solving the heat conduction problems in this scheme the governing equation of heat conduction is firstly written into a system of first order partial differential equations which are integrated over the control volumes around each node point and a finite time step the resulting time integrals are approximated by numerical quadrature with an undetermined weighting parameter the error analysis allows us to determine the parameter and the ratio of the time step to the square of grid spacing by making the error of the scheme as small as possible both theoretical and numerical results show that the proposed high order scheme can achieve at least sixth order accuracy at the expense of the same computing cost as the traditional finite volume method and the high order scheme demonstrates a better performance than the high order compact finite difference method and the runge kutta discontinuous galerkin method

error analysis for a mimetic discretization of the steady stokes problem on polyhedral meshes we present the development convergence analysis and numerical tests of the mimetic finite difference method for the stokes problem on two dimensional polygonal and three dimensional polyhedral meshes

the maximum angle condition for mixed and nonconforming elements application to the stokes equations for the lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition the objective of this paper is to obtain similar results for the raviart thomas interpolation arising in the analysis of mixed methods we prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and moreover that this condition is indeed necessary to have these estimates error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the stokes equations are derived from our results

defect correction based velocity reconstruction for physically consistent simulations of non newtonian flows on unstructured grids abstract a new algorithm to recover centroidal velocities from face normal data on two dimensional unstructured staggered meshes is presented the proposed approach uses iterative defect correction in conjunction with a lower order accurate gauss reconstruction to obtain second order accurate centroidal velocities we derive the conditions that guarantee the second order accuracy of the velocity reconstruction and demonstrate its efficacy on arbitrary polygonal mesh topologies the necessity of the proposed algorithm for non newtonian flow simulations is elucidated through numerical simulations of channel flow driven cavity and backward facing step problems with power law and carreau fluids numerical investigations show that second order accuracy of the reconstructed velocity field is critical to obtaining physically consistent solutions of vorticity dominated flows on non orthogonal meshes it is demonstrated that the spurious solutions are not linked to discrete conservation and arise solely due to the lower order accuracy of velocity reconstruction the importance of the proposed algorithm for hemodynamic simulations is highlighted through studies of laminar flow in an idealized stenosed artery using different blood models

a general reconstruction algorithm for simulating flows with complex 3d immersed boundaries on cartesian grids in the present note a general reconstruction algorithm for simulating incompressible flows with complex immersed boundaries on cartesian grids is presented in the proposed method an arbitrary three dimensional solid surface immersed in the fluid is discretized using an unstructured triangular mesh and all the cartesian grid nodes near the interface are identified then the solution at these nodes is reconstructed via linear interpolation along the local normal to the body in a way that the desired boundary conditions for both pressure and velocity fields are enforced the overall accuracy of the resulting solver is second order as it is demonstrated in two test cases involving laminar flow past a sphere

generalized linear complementarity problems it has been shown by lemke that if a matrix is copositive plus on n then feasibility of the corresponding linear complementarity problem implies solvability in this article we show under suitable conditions that feasibility of ageneralized linear complementarity problem i e defined over a more general closed convex cone in a real hilbert space implies solvability whenever the operator is copositive plus on that cone we show that among all closed convex cones in a finite dimensional real hilbert space polyhedral cones are theonly ones with the property that every copositive plus feasible glcp is solvable we also prove a perturbation result for generalized linear complementarity problems

a decoupled and positivity preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes abstract we suggest a new positivity preserving discrete duality finite volume ddfv scheme for anisotropic diffusion problems on general polygonal meshes like existing ddfv schemes this scheme is built on the primary and dual meshes and has finite volume fv equations for both vertex and cell centered unknowns the transpose of the coefficient matrix for the fv equations of the cell centered unknowns is an m matrix while that for the vertex unknowns is not an m matrix but a symmetric and positive definite matrix by employing a certain truncation technique for the vertex unknowns the positivity preserving property for both categories of unknowns is guaranteed local conservation is strictly maintained for the cell centered unknowns and conditionally maintained for the vertex unknowns since the fv equations of the vertex unknowns can be solved independently the two sets of fv equations are decoupled in contrast to existing nonlinear positivity preserving schemes the new scheme requires no nonlinear iterations for linear problems for nonlinear problems the positivity preserving mechanism of the new scheme is decoupled from its nonlinear iteration so that any nonlinear solver can be adopted moreover the positivity of the discrete solution is proved and the well posedness is analyzed rigorously for linear problems in numerical experiments the new scheme is examined extensively and compared with two positivity preserving cell centered schemes and a nonlinear ddfv scheme numerical results show that the present ddfv scheme achieves second order accuracy and preserves the positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids the high efficiency of the scheme is also demonstrated by the comparison of computation time and number of nonlinear iterations

an overset grid method for 3d unsteady incompressible flows a composite grid numerical method is developed for simulating unsteady three dimensional 3d incompressible flows in complex geometries the governing equations are solved using a second order accurate finite volume method based on the dual time stepping artificial compressibility approach overset grids are employed to discretize arbitrarily complex geometries and a new interface algorithm is developed to facilitate communication between neighboring grids the algorithm is inspired by the necessary and sufficient conditions for satisfying global mass conservation in a composite domain and is simple to implement in 3d numerical experiments show that the new interpolation scheme is superior to straightforward trilinear interpolation of all flow variables as it minimizes non physical spurious oscillations in the overlap region is less sensitive to grid refinement and greatly enhances the computational efficiency of the iterative algorithm the advantages of the new method are especially pronounced when adjacent overset subdomains are discretized with different spatial resolutions the potential of the method as a powerful technique for simulating complex engineering flows is demonstrated by applying it to calculate vortex shedding from a circular cylinder mounted between two endplates and flow in a rectangular channel with two wall mounted obstacles

a direct forcing embedded boundary method with adaptive mesh refinement for fluid structure interaction problems in the present work we developed a structured adaptive mesh refinement s amr strategy for fluid structure interaction problems in laminar and turbulent incompressible flows the computational grid consists of a number of nested grid blocks at different refinement levels the coarsest grid blocks always cover the entire computational domain and local refinement is achieved by the bisection of selected blocks in every coordinate direction the grid topology and data structure is managed using the paramesh toolkit the filtered navier stokes equations for incompressible flow are advanced in time using an explicit second order projection scheme where all spatial derivatives are approximated using second order central differences on a staggered grid for transitional and turbulent flow regimes the large eddy simulation les approach is used where special attention is paid on the discontinuities introduced by the local refinement for all the fluid structure interaction problems reported in this study the complete set of equations governing the dynamics of the flow and the structure are simultaneously advanced in time using a predictor corrector strategy an embedded boundary method is utilized to enforce the boundary conditions on a complex moving body which is not aligned with the grid lines several examples of increasing complexity are given to demonstrate the robustness and accuracy of the proposed formulation

conservation properties of unstructured staggered mesh schemes classic cartesian staggered mesh schemes have a number of attractive properties they do not display spurious pressure modes and they have been shown to locally conserve mass momentum kinetic energy and circulation to machine precision recently a number of generalizations of the staggered mesh approach have been proposed for unstructured triangular or tetrahedral meshes these unstructured staggered mesh methods have been created to retain the attractive pressure aspects and mass conservation properties of the classic cartesian mesh method this work addresses the momentum kinetic energy and circulation conservation properties of unstructured staggered mesh methods it is shown that with certain choices of the velocity interpolation unstructured staggered mesh discretizations of the divergence form of the navier stokes equations can conserve kinetic energy and momentum both locally and globally in addition it is shown that unstructured staggered mesh discretizations of the rotational form of the navier stokes equations can conserve kinetic energy and circulation both locally and globally the analysis includes viscous terms and a generalization of the concept of conservation in the presence of viscosity to include a negative definite dissipation term in the kinetic energy equation these novel conserving unstructured staggered mesh schemes have not been previously analyzed it is shown that they are first order accurate on nonuniform two dimensional unstructured meshes and second order accurate on uniform unstructured meshes numerical confirmation of the conservation properties and the order of accuracy of these unstructured staggered mesh methods is presented

a posteriori error estimation for the discrete duality finite volume discretization of the laplace equation an efficient and fully computable a posteriori error bound is derived for the discrete duality finite volume discretization of the laplace equation on very general two dimensional meshes the main ingredients are the equivalence of this method with a finite element like scheme and tools from the finite element framework numerical tests are performed with a stiff solution on highly nonconforming locally refined meshes and with a singular solution on triangular meshes

natural discretizations for the divergence gradient and curl on logically rectangular grids abstract this is the first in series of papers creating a discrete analog of vector analysis on logically rectangular nonorthogonal nonsmooth grids we introduce notations for 2 d logically rectangular grids describe both cell valued and nodal discretizations for scalar functions and construct the natural discretizations of vector fields using the vector components normal and tangential to the cell boundaries we then define natural discrete analogs of the divergence gradient and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates such as length of elements of coordinate lines areas of elements of coordinate surfaces and elementary volumes we introduce the discrete volume integral of scalar functions the discrete surface integral and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects these theorems include the following the discrete analog of relationship div a 0 if and only if a curl b curl a 0 if and only if a grad if a grad then the line integral does not depend on path and if the line integral of a vector function is equal to zero for any closed path then this vector is the gradient of a scalar function last we define the discrete operators div grad and curl in terms of primitive differencing operators based on forward and backward differences and primitive metric operators related to multiplications of discrete functions by length of edges areas of surfaces and volumes of 3 d cells these formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints

iterative inflow implicit outflow explicit finite volume scheme for level set equations on polyhedron meshes abstract in this paper we propose a cell centered finite volume method for advective and normal flows on polyhedron meshes which is second order accurate in space and time for smooth solutions in order to overcome a time restriction caused by cfl condition an implicit time discretization of inflow fluxes and an explicit time discretization of outflow fluxes are used in an iterative procedure for an efficient computation an 1 ring face neighborhood structure is introduced since it is limited to access unknown variables in an 1 ring face neighborhood structure an iterative procedure is proposed to resolve the limitation of assembled linear system two types of gradient approximations an inflow based gradient and an average based gradient are studied and compared from the point of numerical accuracy numerical schemes are tested for an advective and a normal flow of level set functions illustrating a behavior of the proposed method for an implicit tracking of a smooth and a piecewise smooth interface

a co volume scheme for the rotating shallow water equations on conforming non orthogonal grids a co volume scheme is introduced for the rotating shallow water equations in which both velocity components are specified on cell edges and the thickness variables evolve on both the primary and the dual cell centers the scheme applies to generic conforming and non orthogonal staggered grids including the widely used lat lon quadrilateral grids and the delaunay voronoi tessellations it can be viewed either as coupled c grid schemes on the primary and dual meshes or as an generalization of the traditional e grid scheme on a new non overlapping grid linear dispersive wave analysis shows that the dispersive relations resolved by either the primary or the dual mesh of a uniform quadrilateral staggered grid is the same as those of the z grid scheme the total wavenumber space resolved by the staggered grid is twice as large on which the co volume behaves exactly like the e grid scheme on a uniform hexagon triangular staggered grid the co volume has two steady modes and two inertial gravity modes on the hexagonal mesh and one steady mode two inertial gravity modes and two spurious modes on the triangular mesh on the wavenumber space resolved by either the hexagonal or the triangular mesh the inertial gravity wave modes remain positive and largely monotone for the nonlinear shallow water equations the co volume scheme is shown to preserve the potential vorticity dynamics and the total energy exactly numerical results are presented to corroborate and supplement the analyses

higher order mimetic methods for unstructured meshes a higher order mimetic method for the solution of partial differential equations on unstructured meshes is developed and demonstrated on the problem of conductive heat transfer mimetic discretization methods create discrete versions of the partial differential operators such and the gradient and divergence that are exact in some sense and therefore mimic the important mathematical properties of their continuous counterparts the proposed numerical method is an interesting mixture of both finite volume and finite element ideas while the ideas presented can be applied to arbitrarily high order accuracy we focus in this work on the details of creating a third order accurate method the proposed method is shown to be exact for piecewise quadratic solutions and shows third order convergence on arbitrary triangular tetrahedral meshes the numerical accuracy of the method is confirmed on both two dimensional and three dimensional unstructured meshes the computational cost required for a desired accuracy is analyzed against lower order mimetic methods

convergence analysis of a covolume scheme for maxwell s equations in three dimensions this paper contains error estimates for covolume discretizations of maxwell s equations in three space dimensions several estimates are proved first an estimate for a semi discrete scheme is given second the estimate is extended to cover the classical interlaced time marching technique third some of our unstructured mesh results are specialized to rectangular meshes both uniform and nonuniform by means of some additional analysis it is shown that the spatial convergence rate is one order higher than for the unstructured case

covolume solutions of three dimensional div curl equations the voronoi delaunay mesh systems provide a generalization of the classical rectangular staggered meshes to unstructured meshes in this work it is shown how such covolume discretizations may be applied to div curl systems in three dimensions error estimates are proved and confirmed by a numerical illustration

convergence analysis of fully discrete finite volume methods for maxwell s equations in nonhomogeneous media we will consider both explicit and implicit fully discrete finite volume schemes for solving three dimensional maxwell s equations with discontinuous physical coefficients on general polyhedral domains stability and convergence for both schemes are analyzed we prove that the schemes are second order accurate in time both schemes are proved to be first order accurate in space for the voronoi delaunay grids and second order accurate for nonuniform rectangular grids we also derive explicit expressions for the dependence on the physical parameters in all estimates

a hybrid cartesian immersed boundary method for simulating flows with 3d geometrically complex moving bodies a numerical method is developed for solving the 3d unsteady incompressible navier stokes equations in cartesian domains containing immersed boundaries of arbitrary geometrical complexity moving with prescribed kinematics the governing equations are discretized on a hybrid staggered non staggered grid layout using second order accurate finite difference formulas the discrete equations are integrated in time via a second order accurate dual time stepping artificial compressibility iteration scheme unstructured triangular meshes are employed to discretize complex immersed boundaries the nodes of the surface mesh constitute a set of lagrangian control points used to track the motion of the flexible body at every instant in time the influence of the body on the flow is accounted for by applying boundary conditions at cartesian grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface grid convergence tests are carried out for the flow induced by an oscillating sphere in a cubic cavity which show that the method is second order accurate the method is validated by applying it to calculate flow in a cartesian domain containing a rigid sphere rotating at constant angular velocity as well as flow induced by a flapping wing the ability of the method to simulate flows in domains with arbitrarily complex moving bodies is demonstrated by applying to simulate flow past an undulating fish like body and flow past an anatomically realistic planktonic copepod performing an escape like maneuver euver

unstructured cartesian refinement with sharp interface immersed boundary method for 3d unsteady incompressible flows a novel numerical method is developed for solving the 3d unsteady incompressible navier stokes equations on locally refined fully unstructured cartesian grids in domains with arbitrarily complex immersed boundaries owing to the utilization of the fractional step method on an unstructured cartesian hybrid staggered non staggered grid layout flux mismatch and pressure discontinuity issues are avoided and the divergence free constraint is inherently satisfied to machine zero auxiliary hanging nodes are used to facilitate the discretization of the governing equations the second order accuracy of the solver is ensured by using multi dimension lagrange interpolation operators and appropriate differencing schemes at the interface of regions with different levels of refinement the sharp interface immersed boundary method is augmented with local near boundary refinement to handle arbitrarily complex boundaries the discrete momentum equation is solved with the matrix free newton krylov method and the krylov subspace method is employed to solve the poisson equation the second order accuracy of the proposed method on unstructured cartesian grids is demonstrated by solving the poisson equation with a known analytical solution a number of three dimensional laminar flow simulations of increasing complexity illustrate the ability of the method to handle flows across a range of reynolds numbers and flow regimes laminar steady and unsteady flows past a sphere and the oblique vortex shedding from a circular cylinder mounted between two end walls demonstrate the accuracy the efficiency and the smooth transition of scales and coherent structures across refinement levels large eddy simulation les past a miniature wind turbine rotor parameterized using the actuator line approach indicates the ability of the fully unstructured solver to simulate complex turbulent flows finally a geometry resolving les of turbulent flow past a complete hydrokinetic turbine illustrates the potential of the method to simulate turbulent flows past geometrically complex bodies on locally refined meshes in all the cases the results are found to be in very good agreement with published data and savings in computational resources are achieved

a discrete calculus analysis of the keller box scheme and a generalization of the method to arbitrary meshes abstract the keller box scheme is a face based method for solving partial differential equations that has numerous attractive mathematical and physical properties it is shown that these attractive properties collectively follow from the fact that the scheme discretizes partial derivatives exactly and only makes approximations in the algebraic constitutive relations appearing in the pde the exact discrete calculus associated with the keller box scheme is shown to be fundamentally different from all other mimetic physics capturing numerical methods this suggests that a unique exact discrete calculus does not exist it also suggests that existing analysis techniques based on concepts in algebraic topology in particular the discrete de rham complex are unnecessarily narrowly focused since they do not capture the keller box scheme the discrete calculus analysis allows a generalization of the keller box scheme to non simplectic meshes to be constructed analysis and tests of the method on the unsteady advection diffusion equations are presented

a weak galerkin least squares finite element method for div curl systems abstract in this paper we introduce a weak galerkin least squares method for solving div curl problem this finite element method leads to a symmetric positive definite system and has the flexibility to work with general meshes such as hybrid mesh polytopal mesh and mesh with hanging nodes error estimates of the finite element solution are derived the numerical examples demonstrate the robustness and flexibility of the proposed method

a complementarity problem in mathematical programming in banach space abstract an existence and uniqueness theorem for the nonlinear complementarity problem over arbitrary closed convex cones in a reflexive real banach space is established

multi block large eddy simulations of turbulent boundary layers time developing turbulent boundary layers over an isothermal flat plate at free stream mach numbers of 0 3 and 0 7 are computed using an explicit finite difference method on structured multi block grids the size of each block is adjusted depending on the dimension of the largest structures present locally in the flow this alleviates the cost of calculations in which the wall layer is resolved and may result in substantial savings of memory and cpu time if several layers are used in the calculations presented the near wall region is computed using a domain with a spanwise length l o 820 which is sufficient to contain several streaks this grid block is repeated periodically in the spanwise direction the outer layer which contains larger structures is computed using a domain that is twice as wide l o 1640 although the flow at the interface between the blocks has a periodicity length determined by the inner layer block within a few grid points longer wavelengths are generated the velocity statistics and rms intensities compare well with single block calculations that use substantially more grid points

analysis of a fractional step method on overset grids a fractional step method for solving the incompressible navier stokes equations on overset grids is derived as a matrix factorization of the spatially and temporally discretized system of equations the algorithm is applied to several test problems using second order accurate finite volume flux differencing on staggered grid systems and a hybrid implicit explicit time advancement scheme spatial order of accuracy is shown to depend on the behavior of the overset grid overlap during grid refinement the temporal order of accuracy of the time advancement algorithm on a single grid is maintained on the overset grid

finite dimensional approximation for a class of elliptic obstacle problems in this paper we study the finite dimensional approximation for a class of elliptic obstacle problems involving unbounded domains by using the galerkin method the convergence of the approximation is deduced

analysis of piecewise linear approximations to the generalized stokes problem in the velocity stress pressure formulation in this paper we study continuous piecewise linear polynomial approximations to the generalized stokes equations in the velocity stress pressure first order system formulation by using a cell vertex finite volume least squares scheme this method is composed of a direct cell vertex finite volume discretization step and an algebraic least squares step where the least squares procedure is applied after the discretization process is accomplished this combined approach has the advantages of both finite volume and least squares approaches an error estimate in the h1 product norm for continuous piecewise linear approximating functions is derived it is shown that with respect to the order of approximation for h2 regular exact solutions the method exhibits an optimal rate of convergence in the h1 norm for all unknowns velocity stress and pressure

existence results for the nonlinear complementarity problem and applications to nonlinear analysis abstract we produce an existence result for the nonlinear complementarity problem associated to a weakly locally compact convex cone k in a banach space e and to the difference j f of two nonlinear operators we specialize our result to the case where k is a so called galerkin cone j is the duality mapping and f satisfies a suitable growth condition applications to the fixed point theory and to eigenvalue problems conclude the paper

optimal discontinuous galerkin methods for the acoustic wave equation in higher dimensions in this paper we develop and analyze a new class of discontinuous galerkin dg methods for the acoustic wave equation in mixed form traditional mixed finite element fe methods produce energy conserving schemes but these schemes are implicit making the time stepping inefficient standard dg methods give explicit schemes but these approaches are typically dissipative or suboptimally convergent depending on the choice of numerical fluxes our new method can be seen as a compromise between these two kinds of techniques in the way that it is both explicit and energy conserving locally and globally moreover it can be seen as a generalized version of the raviart thomas fe method and the finite volume method stability and convergence of the new method are rigorously analyzed and we have shown that the method is optimally convergent furthermore in order to apply the new method for unbounded domains we apply our new method with the first order absorbing boundary condition the stability of the resulting numerical scheme is analyzed

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

study of discrete duality finite volume schemes for the peaceman model in this paper we are interested in the finite volume approximation of a system describing miscible displacement in porous media this system is made of two coupled equations an anisotropic diffusion equation on the pressure and a convection diffusion dispersion equation on the concentration of invading fluid the anisotropic diffusion operators in both equations require a special care while discretizing by a finite volume method we focus here on the numerical approximation by some discrete duality finite volume methods after the presentation of the scheme we establish some a priori estimates satisfied by the numerical solution and prove existence and uniqueness of the solution to the scheme we show the efficiency of the schemes through numerical experiments

mimetic discretizations for maxwell s equations we have constructed reliable finite difference methods for approximating the solution to maxwell s equations using accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus in discrete form the numerical approximation does not have spurious modes and mimics many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields numerical examples demonstrate the high quality of the method when the medium is strongly discontinuous and for nonorthogonal nonsmooth computational grids

a new approximation technique for div curl systems in this paper we describe an approximation technique for div curl systems based in l where is a domain in 3 we formulate this problem as a general variational problem with different test and trial spaces the analysis requires the verification of an appropriate inf sup condition this results is a very weak formulation where the solution space is l and the data reside in various negative norm spaces subsequently we consider finite element approximations based on this weak formulation the main approach of this paper involves the development of stable pairs of discrete test and trial spaces with this approach we enlarge the test space so that the discrete inf sup condition holds and use a negative norm least squares formulation to reduce to a uniquely solvable linear system this leads to optimal order estimates for problems with minimal regularity which is important since it is possible to construct magnetostatic field problems whose solutions have low sobolev regularity e g h with 0 s 1 2 the resulting algebraic equations are symmetric positive definite and well conditioned a second approach using a smaller test space which adds terms to the form for stabilization will also be mentioned some numerical results are also presented