a stream virtual element formulation of the stokes problem on polygonal meshes in this paper we propose and analyze a novel stream formulation of the virtual element method vem for the solution of the stokes problem the new formulation hinges upon the introduction of a suitable stream function space characterizing the divergence free subspace of discrete velocities and it is equivalent to the velocity pressure inf sup stable mimetic scheme presented in l beirao da veiga et al j comput phys 228 2009 pp 7215 7232 up to a suitable reformulation into the vem framework both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one

finite element approximation of a sixth order nonlinear degenerate parabolic equation we consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation equation here where generically equation here for any given equation here in addition to showing well posedness of our approximation we prove convergence in space dimensions d leq 3 furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed finally some numerical experiments in one and two space dimensions are presented

the virtual element method for discrete fracture network simulations abstract in this work an optimization based approach presented in berrone et al 2013 2014 10 12 for discrete fracture network simulations is coupled with the virtual element method vem for the space discretization of the underlying darcy law the great flexibility of the vem in handling rather general polygonal elements allows in a natural way for an effective description of irregular solutions starting from an arbitrary triangulation which is built independently of the mesh on other fractures only partial conformity is in fact obtained with this approach numerical results performed on several dfn configurations confirm the viability and efficiency of the resulting method

a superconvergent ldg hybridizable galerkin method for second order elliptic problems we identify and study an ldg hybridizable galerkin method which is not an ldg method for second order elliptic problems in several space dimensions with remarkable convergence properties unlike all other known discontinuous galerkin methods using polynomials of degree k 0 for both the potential as well as the flux the order of convergence in l 2 of both unknowns is k 1 moreover both the approximate potential as well as its numerical trace superconverge in l 2 like norms to suitably chosen projections of the potential with order k 2 this allows the application of element by element postprocessing of the approximate solution which provides an approximation of the potential converging with order k 2 in l 2 the method can be thought to be in between the hybridized version of the raviart thomas and that of the brezzi douglas marini mixed methods

bubble stabilization of discontinuous galerkin methods we analyze the stabilizing effect of the introduction of suitable bubble functions in dg formulations for linear second order elliptic problems working for the sake of simplicity on laplace operator in particular we find that the addition of a single b ubble per element can stabilize the non symmetric formulation of baumann oden most commonly used dg methods need the addition of suitable stabilizing terms in order to provide good convergence properties the typical stabilizing procedure consists in the introduction of penalty terms that penalize the jumps across neigh boring elements sometimes in hyperbolic or in convection dominated problems one also use upwind techniques consisting in replacing the average u u 2 on an internal edge with the upwind value that is u or u according with the direction of the wind this however in most cases can be seen again as a jump stabilization 15 13 10 another possible way of stabilizing dg methods consists in the addition of suit able terms this time internal to each element of the so ca lled hughes francatype in general the integral of the original equation or one of t he original equations written in strong form inside each element in terms of the finite element unknowns trial functions multiplied by a similar expression act ing on the test functions the most famous stabilization of this type for standard galerkin methods is surely the supg stabilization of convection dominated equations 12 a typical problem in these cases is the choice of the proper stabilization coefficient to be put in front of the stabilizing term in a recent paper see 6 we pointed out that in dg methods the jumps are themselves to be regarded as equations so that jump stabilizations and hence upwind could be regarded as hughes franca stabilizations as well and indeed the optimal choice of the coefficient in a jump stabilizatio n term is still a subject that might need a further investigation

virtual element methods for plate bending problems abstract we discuss the application of virtual elements to linear plate bending problems in the kirchhoff love formulation as we shall see in the virtual element environment the treatment of the c 1 continuity condition is much easier than for traditional finite elements the main difference consists in the fact that traditional finite elements for every element k and for every given set of degrees of freedom require the use of a space of polynomials or piecewise polynomials for composite elements for which the given set of degrees of freedom is unisolvent for virtual elements instead we only need unisolvence for a space of smooth functions that contains a subset made of polynomials whose degree determines the accuracy as we shall see the non polynomial part of our local spaces does not need to be known in detail and therefore the construction of the local stiffness matrix is simple and can be done for much more general geometries

on the virtual element method for topology optimization on polygonal meshes it is well known that the solution of topology optimization problems may be affected both by the geometric properties of the computational mesh which can steer the minimization process towards local and non physical minima and by the accuracy of the method employed to discretize the underlying differential problem which may not be able to correctly capture the physics of the problem in light of the above remarks in this paper we consider polygonal meshes and employ the virtual element method vem to solve two classes of paradigmatic topology optimization problems one governed by nearly incompressible and compressible linear elasticity and the other by stokes equations several numerical results show the virtues of our polygonal vem based approach with respect to more standard methods

on the virtual element method for three dimensional linear elasticity problems on arbitrary polyhedral meshes abstract we explore the recently proposed virtual element method vem for the numerical solution of boundary value problems on arbitrary polyhedral meshes more specifically we focus on the linear elasticity equations in three dimensions and elaborate upon the key concepts underlying the first order vem while the point of departure is a conforming galerkin framework the distinguishing feature of vem is that it does not require an explicit computation of the trial and test spaces thereby circumventing a barrier to standard finite element discretizations on arbitrary grids at the heart of the method is a particular kinematic decomposition of element deformation states which in turn leads to a corresponding decomposition of strain energy by capturing the energy of linear deformations exactly one can guarantee satisfaction of the patch test and optimal convergence of numerical solutions the decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation as we show computing these projection maps and subsequently the local stiffness matrices in practice reduces to the computation of purely geometric quantities in addition to discussing aspects of implementation of the method we present several numerical studies in order to verify convergence of the vem and evaluate its performance for various types of meshes

adaptive time stepping for vesicle suspensions we present an adaptive high order accurate time stepping numerical scheme for the flow of vesicles suspended in stokesian fluids our scheme can be summarized as an approximate implicit spectral deferred correction sdc method applying a fully implicit sdc scheme to vesicle flows is prohibitively expensive for this reason we introduce several approximations our scheme is based on a semi implicit linearized low order time stepping method our discretization is spectrally accurate in space we expect that the accuracy can be arbitrary order but our examples suffer from order reduction which limits the observed accuracy to third order we also use invariant properties of vesicle flows constant area and boundary length in two dimensions to reduce the computational cost of error estimation for adaptive time stepping we present results in two dimensions for single vesicle flows constricted geometry flows converging flows and flows in a couette apparatus we experimentally demonstrate that the proposed scheme enables automatic selection of the time step size and high order accuracy

the conforming virtual element method for polyharmonic problems abstract in this work we exploit the capability of virtual element methods in accommodating approximation spaces featuring high order continuity to numerically approximate differential problems of the form p u f p 1 more specifically we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems and prove an abstract result that states the convergence of the method in suitable norms

virtual element method for fourth order problems l 2 estimates abstract we analyse the family of c 1 virtual elements introduced in brezzi and marini 2013 for fourth order problems and prove optimal estimates in l 2 and in h 1 via classical duality arguments

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

the nonconforming virtual element method for the stokes equations we present the nonconforming virtual element method vem for the numerical approximation of velocity and pressure in the steady stokes problem the pressure is approximated using discontinuous piecewise polynomials while each component of the velocity is approximated using the nonconforming virtual element space on each mesh element the local virtual space contains the space of polynomials of up to a given degree plus suitable nonpolynomial functions the virtual element functions are implicitly defined as the solution of local poisson problems with polynomial neumann boundary conditions as typical in vem approaches the explicit evaluation of the non polynomial functions is not required this approach makes it possible to construct nonconforming virtual spaces for any polynomial degree regardless of the parity for two and three dimensional problems and for meshes with very general polygonal and polyhedral elements we show that the nonconforming vem is inf sup stable and establish optimal a pri

a c 1 virtual element method for the cahn hilliard equation with polygonal meshes in this paper we develop an evolution of the c 1 virtual elements of minimal degree for the approximation of the cahn hilliard equation the proposed method has the advantage of being conforming in h 2 and making use of a very simple set of degrees of freedom namely 3 degrees of freedom per vertex of the mesh moreover although the present method is new also on triangles it can make use of general polygonal meshes as a theoretical and practical support we prove the convergence of the semidiscrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests

conforming polygonal finite elements summary in this paper conforming finite elements on polygon meshes are developed polygonal finite elements provide greater flexibility in mesh generation and are better suited for applications in solid mechanics which involve a significant change in the topology of the material domain in this study recent advances in meshfree approximations computational geometry and computer graphics are used to construct different trial and test approximations on polygonal elements a particular and notable contribution is the use of meshfree natural neighbour nn basis functions on a canonical element combined with an affine map to construct conforming approximations on convex polygons this numerical formulation enables the construction of conforming approximation on n gons n 3 and hence extends the potential applications of finite elements to convex polygons of arbitrary order numerical experiments on second order elliptic boundary value problems are presented to demonstrate the accuracy and convergence of the proposed method copyright 2004 john wiley sons ltd

fundamental solutions method for elliptic boundary value problems we consider a procedure for solving boundary value problems for elliptic homogeneous equations known as the fundamental solutions method we prove its applicability for some second order operators as well as for fourth order ones the boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest in a specific case of the laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions numerical examples are given among which are simple signorini s problems for harmonic and lame operators

spectrally accurate quadratures for evaluation of layer potentials close to the boundary for the 2d stokes and laplace equations dense particulate flow simulations using integral equation methods demand accurate evaluation of stokes layer potentials on arbitrarily close interfaces in this paper we generalize techniques for close evaluation of laplace double layer potentials in j helsing and r ojala j comput phys 227 2008 pp 2899 2921 we create a globally compensated trapezoid rule quadrature for the laplace single layer potential on the interior and exterior of smooth curves this exploits a complex representation a product quadrature in the style of kress for the sawtooth function careful attention to branch cuts and second kind barycentric type formulae for cauchy integrals and their derivatives upon this we build accurate single and double layer stokes potential evaluators by expressing them in terms of laplace potentials we test their convergence for vesicle vesicle interactions for an extensive set of laplace and stokes problems and when applying the system matrix in a boundary value problem solver

spectral ewald acceleration of stokesian dynamics for polydisperse suspensions in this work we develop the spectral ewald accelerated stokesian dynamics seasd a novel computational method for dynamic simulations of polydisperse colloidal suspensions with full hydrodynamic interactions seasd is based on the framework of stokesian dynamics sd with extension to compressible solvents and uses the spectral ewald se method lindbo and tornberg 2010 29 for the wave space mobility computation to meet the performance requirement of dynamic simulations we use graphic processing units gpu to evaluate the suspension mobility and achieve an order of magnitude speedup compared to a cpu implementation for further speedup we develop a novel far field block diagonal preconditioner to reduce the far field evaluations in the iterative solver and seasd nf a polydisperse extension of the mean field brownian approximation of banchio and brady 2003 39 we extensively discuss implementation and parameter selection strategies in seasd and demonstrate the spectral accuracy in the mobility evaluation and the overall o n log n computation scaling we present three computational examples to further validate seasd and seasd nf in monodisperse and bidisperse suspensions the short time transport properties the equilibrium osmotic pressure and viscoelastic moduli and the steady shear brownian rheology our validation results show that the agreement between seasd and seasd nf is satisfactory over a wide range of parameters and also provide significant insight into the dynamics of polydisperse colloidal suspensions

a new discretization for mth laplace equations with arbitrary polynomial degrees this paper introduces new mixed formulations and discretizations for m th laplace equations of the form 1 m delta m u f for arbitrary m 1 2 3 dots based on novel helmholtz type decompositions for tensor valued functions the new discretizations allow for ansatz spaces of arbitrary polynomial degree and the lowest order choice coincides with the nonconforming fems of crouzeix and raviart for m 1 and of morley for m 2 since the derivatives are directly approximated the lowest order discretizations consist of piecewise affine and piecewise constant functions for any m 1 2 dots moreover a uniform implementation for arbitrary m is possible besides the a priori and a posteriori analysis this paper proves optimal convergence rates for adaptive algorithms for the new discretizations

a posteriori error estimates for the virtual element method an posteriori error analysis for the virtual element method vem applied to general elliptic problems is presented the resulting error estimator is of residual type and applies on very general polygonal polyhedral meshes the estimator is fully computable as it relies only on quantities available from the vem solution namely its degrees of freedom and element wise polynomial projection upper and lower bounds of the error estimator with respect to the vem approximation error are proven the error estimator is used to drive adaptive mesh refinement in a number of test problems mesh adaptation is particularly simple to implement since elements with consecutive co planar edges faces are allowed and therefore locally adapted meshes do not require any local mesh post processing

a fast algorithm for simulating multiphase flows through periodic geometries of arbitrary shape this paper presents a new boundary integral equation bie method for simulating particulate and multiphase flows through periodic channels of arbitrary smooth shape in two dimensions the authors consider a particular system multiple vesicles suspended in a periodic channel of arbitrary shape to describe the numerical method and test its performance rather than relying on the periodic green s function as classical bie methods do the method combines the free space green s function with a small auxiliary basis and imposes periodicity as an extra linear condition as a result we can exploit existing free space solver libraries quadratures and fast algorithms and handle a large number of vesicles in a geometrically complex channel spectral accuracy in space is achieved using the periodic trapezoid rule and product quadratures while a first order semi implicit scheme evolves particles by treating the vesicle channel interactions explicitly new constraint correction formulas are introduced that pre

an analysis of convex relaxations for map estimation of discrete mrfs the problem of obtaining the maximum a posteriori estimate of a general discrete markov random field i e a markov random field defined using a discrete set of labels is known to be np hard however due to its central importance in many applications several approximation algorithms have been proposed in the literature in this paper we present an analysis of three such algorithms based on convex relaxations i lp s the linear programming lp relaxation proposed by schlesinger 1976 for a special case and independently in chekuri et al 2001 koster et al 1998 and wainwright et al 2005 for the general case ii qp rl the quadratic programming qp relaxation of ravikumar and lafferty 2006 and iii socp ms the second order cone programming socp relaxation first proposed by muramatsu and suzuki 2003 for two label problems and later extended by kumar et al 2006 for a general label set we show that the socp ms and the qp rl relaxations are equivalent furthermore we prove that despite the flexibility in the form of the constraints objective function offered by qp and socp the lp s relaxation strictly dominates i e provides a better approximation than qp rl and socp ms we generalize these results by defining a large class of socp and equivalent qp relaxations which is dominated by the lp s relaxation based on these results we propose some novel socp relaxations which define constraints using random variables that form cycles or cliques in the graphical model representation of the random field using some examples we show that the new socp relaxations strictly dominate the previous approaches

equivalent projectors for virtual element methods in the original virtual element space with degree of accuracy k projector operators in the h 1 seminorm onto polynomials of degree k can be easily computed on the other hand projections in the l 2 norm are available only on polynomials of degree k 2 directly from the degrees of freedom here we present a variant of the virtual element method that allows the exact computations of the l 2 projections on all polynomials of degree k the interest of this construction is illustrated with some simple examples including the construction of three dimensional virtual elements the treatment of lower order terms the treatment of the right hand side and the l 2 error estimates

an accurate integral equation method for simulating multi phase stokes flow we introduce a numerical method based on an integral equation formulation for simulating drops in viscous fluids in the plane it builds upon the method introduced by kropinski in 2001 17 but improves on it by adding an interpolatory quadrature approach for handling near singular integrals such integrals typically arise when drop boundaries come close to one another and are difficult to compute accurately using standard quadrature rules adapting the interpolatory quadrature method introduced by helsing and ojala in 2008 11 to the current application very general drop configurations can be handled while still maintaining stability and high accuracy the performance of the new method is demonstrated by some challenging numerical examples