a semi implicit finite element method for viscous lipid membranes a finite element formulation to approximate the behavior of lipid membranes is proposed the mathematical model incorporates tangential viscous stresses and bending elastic forces together with the inextensibility constraint and the enclosed volume constraint the membrane is discretized by a surface mesh made up of planar triangles over which a mixed formulation velocity curvature is built based on the viscous bilinear form boussinesq scriven operator and the laplace beltrami identity relating position and curvature a semi implicit approach is then used to discretize in time with piecewise linear interpolants for all variables two stabilization terms are needed the first one stabilizes the inextensibility constraint by a pressure gradient projection scheme codina and blasco 1997 33 the second couples curvature and velocity to improve temporal stability as proposed by bansch 2001 36 the volume constraint is handled by a lagrange multiplier which turns out to be the internal pressure and an analogous strategy is used to filter out rigid body motions the nodal positions are updated in a lagrangian manner according to the velocity solution at each time step an automatic remeshing strategy maintains suitable refinement and mesh quality throughout the simulation numerical experiments show the convergent and robust behavior of the proposed method stability limits are obtained from numerous relaxation tests and convergence with mesh refinement is confirmed both in the relaxation transient and in the final equilibrium shape virtual tweezing experiments are also reported computing the dependence of the deformed membrane shape with the tweezing velocity a purely dynamical effect for sufficiently high velocities a tether develops which shows good agreement both in its final radius and in its transient behavior with available analytical solutions finally simulation results of a membrane subject to the simultaneous action of six tweezers illustrate the robustness of the method

on the vanishing viscosity limit for the 3d navier stokes equations with a slip boundary condition where and below and denote the div and curl operators respectively n is the outward normal and is the unit tangential vector of the investigation of vanishing viscosity limit of solutions of the navier stokes equations both in the two and three spacial dimensional cases is a classical issue there are two related questions arising from here one is how to describe the inviscid limiting behavior of the navier stokes equation and the other is that does the euler equation can be approximated by the navier stokes equations in the case that the solution to the this research is supported in part by zheng ge ru foundation and hong kong rgc earmarked research grants cuhk 4028 04p cuhk 4040 02p and cuhk 4279 00p

geometrically consistent mesh modification a new paradigm of adaptivity is to execute refinement coarsening and smoothing of meshes on manifolds with incomplete information about their geometry and yet preserve position and curvature accuracy we refer to this collectively as geometrically consistent gc mesh modification we discuss the concept of discrete gc show the failure of naive approaches and propose and analyze a simple algorithm that is gc and accuracy preserving

localizations on complex networks we study the structural characteristics of complex networks using the representative eigenvectors of the adjacent matrix the probability distribution function of the components of the representative eigenvectors are proposed to describe the localization on networks where the euclidean distance is invalid several quantities are used to describe the localization properties of the representative states such as the participation ratio the structural entropy and the probability distribution function of the nearest neighbor level spacings for spectra of complex networks whole cell networks in the real world and the watts strogatz small world and barabasi albert scale free networks are considered the networks have nontrivial localization properties due to the nontrivial topological structures it is found that the ascending order ranked series of the occurrence probabilities at the nodes behave generally multifractally this characteristic can be used as a structural measure of complex networks

some properties of wkb series we investigate some properties of the wkb series for arbitrary analytic potentials and then specifically for potentials xn n even where more explicit formulae for the wkb terms are derived our main new results are i we find the explicit functional form for the general wkb terms k where one has only to solve a general recursion relation for the rational coefficients ii we give a systematic algorithm for an essential simplification of the integrated wkb terms k dx that enter the energy eigenvalue equation and iii we derive almost explicit formulae for the wkb terms for the energy eigenvalues of the homogeneous power law potentials v x xn where n is even

study of coating flow by the finite element method abstract silliman s analysis of slot coating is extended to accommodate film flows with highly bent menisci as in slide and curtain coating by combining polar and cartesian coordinate parametrizations of meniscus shape the nonlinear algebraic equations from the subdomain galerkin weighted residual method and finite element basis functions are solved by newton s method which is perfected for the free boundary problem convergence behavior is examined slot coating results reveal slow flow zones and show that at high capillary number and low metering rate the downstream meniscus can no longer attach to the slot comparison of coyne and elrod s approximate solutions with the finite element solutions shows that their assumption of a parabolic velocity profile leads to reasonable approximations of meniscus shapes

geometric regularity criteria for incompressible navier stokes equations with navier boundary conditions abstract we study the regularity criteria for weak solutions to the 3d incompressible navier stokes equations in terms of the direction of vorticity taking into account the boundary conditions a boundary regularity theorem is proved on regular curvilinear domains with a family of oblique derivative boundary conditions provided that the directions of vorticity are coherently aligned up to the boundary as an application we establish the boundary regularity for weak solutions to navier stokes equations in round balls half spaces and right circular cylindrical ducts subject to the classical navier and kinematic boundary conditions

a posteriori error analysis for the mean curvature flow of graphs we study the equation describing the motion of a nonparametric surface according to its mean curvature flow this is a nonlinear nonuniformly parabolic pde that can be discretized in space via a finite element method we conduct an aposteriori error analysis of the spatial discretization and derive upper bounds on the error in terms of computable estimators based on local residual indicators the reliability of the estimators is illustrated with two numerical simulations one of which treats the case of a singular solution

current flow in random resistor networks the role of percolation in weak and strong disorder we study the current flow paths between two edges in a random resistor network on a ltimes l square lattice each resistor has resistance e ax where x is a uniformly distributed random variable and a controls the broadness of the distribution we find a the scaled variable uequiv l a nu where nu is the percolation connectedness exponent fully determines the distribution of the current path length ell for all values of u for ugg 1 the behavior corresponds to the weak disorder limit and ell scales as ellsim l while for ull 1 the behavior corresponds to the strong disorder limit with ellsim l dscriptsize opt where dscriptsize opt 1 22pm0 01 is the optimal path exponent b in the weak disorder regime there is a length scale xisim a nu below which strong disorder and critical percolation characterize the current path

variational formulations for surface tension capillarity and wetting the interest in the simulation of flows with significant surface tension effects has grown significantly in recent years this has been driven by the substantial advances made in the measurement and manufacturing of microscopic systems since at small length scales surface phenomena are dominant in this article surface tension capillarity and wetting effects are discussed in terms of the virtual work principle and shape sensitivity starting from first principles and arriving at variational formulations that are adequate for numerical treatment by finite elements for example to make the exposition self contained some elements of differential geometry are recalled using a formulation that is fully in cartesian coordinates and may thus be more friendly to readers not familiar with covariant derivatives all necessary results are proved in this cartesian formulation several numerical examples computed with a finite element level set formulation are used to illustrate this challenging physical problem

some numerical methods for the computation of capillary free boundaries governed by the navier stokes equations in the trial method newton s method and the total linearization method are applied to a simple one dimensional free boundary fb problem for which the solution is known attention will be focused on convergence speed and computer implementation facilities of these methods section 2 discusses static fb problems in two and three dimensional containers these fbs are governed by the laplace young equation in 3 two stationary fb problems governed by the navier stokes equations are formulated and these problems are solved using the three methods introduced in 1 a thermocapillary fb problem governed by the navier stokes equations coupled with the heat equation is studied in 4 in 5 a moving fb problem governed by the navier stokes equations will be considered within the context of perturbation theory finally in 6 some remarks are made on present and future research in the field of capillary fb problems governed by the navier stokes equations

finite element discretization of the navier stokes equations with a free capillary surface the instationary navier stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions a stable finite element discretization is presented the key idea is the treatment of the curvature terms by a variational formulation in the context of a discontinuous in time space time element discretization stability in weak energy norms can be proved numerical examples in 2 and 3 space dimensions are given

numerical assessment of stability of interface discontinuous finite element pressure spaces abstract the stability of two recently developed pressure spaces has been assessed numerically the space proposed by ausas et al r f ausas f s sousa g c buscaglia an improved finite element space for discontinuous pressures comput methods appl mech engrg 199 2010 1019 1031 which is capable of representing discontinuous pressures and the space proposed by coppola owen and codina a h coppola owen r codina improving eulerian two phase flow finite element approximation with discontinuous gradient pressure shape functions int j numer methods fluids 49 2005 1287 1304 which can represent discontinuities in pressure gradients we assess the stability of these spaces by numerically computing the inf sup constants of several meshes the inf sup constant results as the solution of a generalized eigenvalue problems both spaces are in this way confirmed to be stable in their original form an application of the same numerical assessment tool to the stabilized equal order p 1 p 1 formulation is then reported an interesting finding is that the stabilization coefficient can be safely set to zero in an arbitrary band of elements without compromising the formulation s stability an analogous result is also reported for the mini element p 1 p 1 when the velocity bubbles are removed in an arbitrary band of elements

time discrete higher order ale formulations stability arbitrary lagrangian eulerian ale formulations deal with pdes on deformable domains upon extending the domain velocity from the boundary into the bulk with the purpose of keeping mesh regularity this arbitrary extension has no effect on the stability of the pde but may influence that of a discrete scheme we examine this critical issue for higher order time stepping without space discretization we propose time discrete discontinuous galerkin dg numerical schemes of any order for a time dependent advection diffusion model problem in moving domains and study their stability properties the analysis hinges on the validity of the reynolds identity for dg exploiting the variational structure and assuming exact integration we prove that our conservative and nonconservative dg schemes are equivalent and unconditionally stable the same results remain true for piecewise polynomial ale maps of any degree and suitable quadrature that guarantees the validity of the reynolds identity this approach general

semiclassical analysis of wigner functions in this work we study the wigner functions which are the quantum analogues of the classical phase space density and show how a full rigorous semiclassical scheme for all orders of can be constructed for them without referring to the actual coordinate space wavefunctions from which the wigner functions are typically calculated we find such a picture by a careful analysis around the stationary points of the main quantization equation and apply this approach to the harmonic oscillator solving it for all orders of

numerical treatment of the navier stokes equations with slip boundary condition in this article we present efficient numerical methods for the navier stokes equations with slip boundary conditions a first method is based on a saddle point formulation of the slip boundary condition a congruent gradient cg method is applied to the schur complement operator in order to solve the problem we present two preconditioners for the cg method which result in convergence rates independent of the mesh size for a second method the slip boundary condition is enforced pointwise for nodal values of the velocity at boundary nodes

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

optimal paths in disordered media scaling of the crossover from self similar to self affine behavior we study optimal paths in disordered energy landscapes using energy distributions of the type p mathrm log 10 e mathrm const that lead to the strong disorder limit if we truncate the distribution so that p mathrm log 10 e mathrm const only for e mathrm min l el e mathrm max and p mathrm log 10 e 0 otherwise we obtain a crossover from self similar strong disorder to self affine moderate disorder behavior at a path length mathcal l ifmmode times else texttimes fi we find that mathcal l ifmmode times else texttimes fi ensuremath propto mathrm log 10 e mathrm max e mathrm min ensuremath kappa where the exponent ensuremath kappa has the value ensuremath kappa 1 60 ifmmode pm else textpm fi 0 03 both in d 2 and d 3 we show how the crossover can be understood from the distribution of local energies on the optimal paths

a hybrid level set front tracking approach for finite element simulations of two phase flows abstract in this paper we give details on the numerical realization of a new finite element method for the simulation of two phase flows which was recently introduced in basting and weismann 2013 the main ingredient is a hybrid representation of the interface between the fluid phases an implicit description of the interface is given by a level set function and an explicit representation is obtained from aligning edges of the computational mesh to the implicitly described interface this step is done by a black box optimization based mesh smoothing approach which does not change the topology of the mesh while guaranteeing optimal mesh quality furthermore we make use of quadratic isoparametric elements to increase the approximation quality of the discrete interface due to the alignment discontinuities of the solution variables pressure can be captured accurately while a variational treatment of the curvature allows for a precise approximation of surface tension we present our time discretization scheme for the coupled navier stokes level set equations and discuss our space discretization based on the so called subspace projection method spm to account for discontinuities across the interface we present two numerical examples for which reference solutions exist we consider the oscillation of a single droplet and provide our results for an established two phase flow benchmark problem

on the parametric finite element approximation of evolving hypersurfaces in r 3 we present a variational formulation of motion by minus the laplacian of curvature and mean curvature flow as well as related second and fourth order flows of a closed hypersurface in r 3 on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches the presented scheme has very good properties with respect to the distribution of mesh points and if applicable volume conservation

variational formulation of a model free boundary problem the purpose of this work is to present an error analysis of the numerical approximation by a finite element method of a free surface problem the analysis has been done in an abstract model which has many of the features of a free surface problem for a viscous liquid we study in this paper how the numerical approximation of the free boundary affects the approximation of the other variables of the problem and vice versa we present the numerical analysis of a free boundary problem that is intended to incorporate many of the difficulties found in a class of models of fluid flow phenomena with free surfaces one such phenomenon which motivates the current work is the flow of a liquid constrained only partly by a container that is in which a part of the boundary of the domain filled by the liquid is an interface with another liquid of much smaller density and for which surface tension plays a significant role in determining the shape of the free surface one model for the behavior of such liquids is based on the assumption that the surface tension between the two liquids is proportional to the curvature of the free surface the constant of proportionality is a physical property of the two fluids this model has been studied extensively in recently years both experimentally cf jean and pritchard 15 and pritchard 19 theoretically cf allain 2 beale 4 bemelmans 5 jean 14 pukhnachov 20 and solonnikov 27 asymptotically keller and miksis 16 and computationally cf cuvelier 10 ryskin and leal 23 and saito and scriven 24 our purpose here is to establish a framework for the analysis of convergence properties of the computational techniques being used the only previous work that we are aware of in this direction is by nitsche 18 in the first section of the paper we define our model problem in classical terms in the second section we construct a variational formulation for the problem that has two new features one is that it allows the existence of a solution to be proved with weaker assumptions on the data than has been possible received february 26 1990 revised november 20 1990 1980 mathematics subject classification 1985 revision primary 65n30

finite element discretization error analysis of a general interfacial stress functional the finite element discretization of a variable interfacial stress tensor sigma gamma x is considered variable interfacial tension is included as a special case interfacial stress for example occurs in two phase flow problems in the weak formulation the interfacial stress gives rise to a functional which is supported on the interface gamma a finite element discretization of this functional is presented and analyzed the discretization admits almost independent meshes for the approximation of the interface and the approximation of the flow variables the main result is an mathcal o h k 1 2 error bound for the interfacial stress functional in a natural norm if the discrete interface is an mathcal o h k 1 approximation of gamma the bound is shown to be sharp in numerical experiments

transport and percolation theory in weighted networks we study the distribution p of the equivalent conductance for erdos renyi er and scale free sf weighted resistor networks with n nodes each link has conductance geax where x is a random number taken from a uniform distribution between 0 and 1 and the parameter a represents the strength of the disorder we provide an iterative fast algorithm to obtain p and compare it with the traditional algorithm of solving kirchhoff equations we find both analytically and numerically that p for er networks exhibits two regimes i a low conductance regime for eapc where pc 1 k is the critical percolation threshold of the network and k is the average degree of the network in this regime p is independent of n and follows the power law p where 1 k a ii a high conductance regime for eapc in which we find that p has strong n dependence and scales as pf apc n 1 3 for sf networks with degree distribution pkk kminkkmax we find numerically also two regimes similar to those found for er networks

an algorithm for evolutionary surfaces an algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of poisson problems on a series of surfaces this gives a new method to solve plateau s problem forh surfaces

a capillary free boundary problem governed by the navier stokes equations abstract in this paper an unsteady capillary free boundary in a viscous imcompressible fluid is studied from the numerical point of view the linearized problem is formulated and existence and uniqueness of a solution is proved the linearized problem is discretized by a finite element method in space and a finite difference method in time the numerical scheme is proved to be convergent and stable the method is applied to the computation of free boundaries eigenfrequencies damping factors and vibration modes the influence of the initial conditions the boundary conditions the contact angle the ohnesorge number and the bond number is studied

a variational shape optimization approach for image segmentation with a mumford shah functional we introduce a novel computational method for a mumford shah functional which decomposes a given image into smooth regions separated by closed curves casting this as a shape optimization problem we develop a gradient descent approach at the continuous level that yields nonlinear pde flows we propose time discretizations that linearize the problem and space discretization by continuous piecewise linear finite elements the method incorporates topological changes such as splitting and merging for detection of multiple objects space time adaptivity and a coarse to fine approach to process large images efficiently we present several simulations that illustrate the performance of the method and investigate the model sensitivity to various parameters

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

scaling of optimal path lengths distribution in complex networks we study the distribution of optimal path lengths in random graphs with random weights associated with each link disorder with each link i we associate a weight i expari where ri is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we suggest in an analogy with the average length of the optimal path that the distribution of optimal path lengths has a universal form that is controlled by the expression 1 pc a where is the optimal path length in strong disorder a and pc is the percolation threshold this relation is supported by numerical simulations for erd s renyi and scale free graphs we explain this phenomenon by showing explicitly the transition between strong disorder and weak disorder at different length scales in a single network

universal behavior of optimal paths in weighted networks with general disorder we study the statistics of the optimal path in both random and scale free networks where weights are taken from a general distribution p w we find that different types of disorder lead to the same universal behavior specifically we find that a single parameter s defined as al 1 v for d dimensional lattices and s defined as an 1 3 for random networks determines the distributions of the optimal path length including both strong and weak disorder regimes here v is the percolation connectivity exponent and a depends on the percolation threshold and p w we show that for a uniform p w poisson or gaussian the crossover from weak to strong does not occur and only weak disorder exists

an asymptotic solution approach for elliptic equations with discontinuous coefficients when the coefficients of an elliptic problem have jumps of several orders of magnitude across an embedded interface many iterative solvers exhibit deteriorated convergence properties or a loss of efficiency and it is difficult to achieve high solution accuracies in the whole domain in this paper we present an asymptotic solution approach for the elliptic problem b x u x f x on a domain w w w with piecewise constant coefficients b b with b b and prescribed jump conditions at an embedded interface c separating the domains w and w we are in particular focusing on a problem related to fluid mechanics namely incompressible two phase flow with a large density ratio across the phase boundary where an accurate solution of the velocity depends on the accurate solution of a pressure poisson equation with equal local relative errors in the whole domain instead of solving the equation in a single solution step we decompose the problem into two consecutive problems based on an asymptotic analysis of the physical problem where each problem is asymptotically independent of the ratio of coefficients e b b the proposed methods lead to a robust and accurate solution of the elliptic problem using standard black box iterative solvers

epidemic spreading with nonlinear infectivity in weighted scale free networks abstract in this paper we investigate the epidemic spreading for the sir model in weighted scale free networks with the nonlinear infectivity and weighted transmission rate concretely we introduce the infectivity exponent and the weight exponent into the epidemic system then examine the impact of and on the epidemic spreading we show that one can adjust the values of and to rebuild a nonzero finite epidemic threshold furthermore we also find the infectivity exponent has a stronger effect not only on the epidemic threshold but also on the epidemic prevalence in addition it is also interesting to see that the absence of the epidemic threshold appears not very dejected since the prevalence grows much more slowly as the transmission rate increases

exact wkb expansions for some potentials we find wkb expansions for the spectra of the three dimensional isotropic harmonic oscillator and eckart and morse potentials it is shown that for all these potentials the series are convergent and when summed yield the exact energy spectra these results support our working hypothesis that the wkb series for the energy spectra are convergent for all one dimensional solvable potentials and their sums yield the exact energies

finite element method for time dependent incompressible free surface flow we present a finite element method for time dependent incompressible free surface fluid flow problems described by the navier stokes equations the elements chosen have dimensions in both space and time and the resulting system of equations is block tridiagonal and lends itself to solution by standard techniques in the present article we restrict our attention to two dimensional problems although three dimensional problems may be solved by a straightforward generalization the method is essentially an implicit time stepping technique and therefore stable even for relatively large time steps with this choice of elements the method is completely adaptive to the changing nature of the solution an iterative procedure is used to find the position of the free surface this procedure is found to be rapidly convergent determining accurately the shape of the free surface within a few iterations numerical results are given for the problem of entrainment of fluid by a vertically moving plate which has applications to the chemical engineering problems of the free coating of metals we also consider the problem of circulation flow in a rectangular channel

stability analysis of pressure correction schemes for the navier stokes equations with traction boundary conditions abstract we present a stability analysis for two different rotational pressure correction schemes with open and traction boundary conditions first we provide a stability analysis for a rotational version of the grad div stabilized scheme of bonito et al 2015 this scheme turns out to be unconditionally stable provided the stabilization parameter is suitably chosen we also establish a conditional stability result for the boundary correction scheme presented in bansch 2014 these results are shown by employing the equivalence between stabilized gauge uzawa methods and rotational pressure correction schemes with traction boundary conditions

a stable numerical method for the dynamics of fluidic membranes we develop a finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in navier stokes flow local inextensibility of the membrane is ensured by solving a tangential navier stokes equation taking surface viscosity effects of boussinesq scriven type into account in our approach the bulk and surface degrees of freedom are discretized independently which leads to an unfitted finite element approximation of the underlying free boundary problem bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by dziuk numer math 111 55 80 2008 the obtained numerical scheme can be shown to be stable and to have good mesh properties finally the evolution of membrane shapes is studied numerically in different flow situations in two and three space dimensions the numerical results demonstrate the robustness of the method and it is observed that the conservation properties are fulfilled to a high precision

effect of disorder strength on optimal paths in complex networks we study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path opt in a disordered erd s renyi er random network and scale free sf network each link i is associated with a weight ti expsarid where ri is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we find that for any finitea there is a crossover network size n sad at which the transition occurs for n n sad the scaling behavior of opt is in the strong disorder regime with opt n 1 3 for er networks and for sf networks with lo 4 and opt n sl 3d sl 1d for sf networks with 3 l 4 for n n sad the scaling behavior is in the weak disorder regime with opt ln n for er networks and sf networks with l 3 in order to study the transition we propose a measure which indicates how close or far the disordered network is from the limit of strong disorder we propose a scaling ansatz for this measure and demonstrate its validity we proceed to derive the scaling relation between n sad and a we find thatn sa d a 3 for er networks and for sf networks with lo 4 and n sa d a sl 1d sl 3d for sf networks with 3 l 4

droplet footprint control controlling droplet shape via surface tension has numerous technological applications such as droplet lenses and lab on a chip this motivates a partial differential equation constrained shape optimization approach for controlling the shape of droplets on flat substrates by controlling the surface tension of the substrate we use shape differential calculus to derive an l 2 gradient flow approach to compute equilibrium shapes for sessile droplets on substrates we then develop a gradient based optimization method to find the substrate surface tension coefficient yielding an equilibrium droplet shape with a desired footprint i e the liquid solid interface has a desired shape moreover we prove a sensitivity result with respect to the substrate surface tensions for the free boundary problem associated with the footprint numerical results are also presented to showcase the method

a subspace projection method for the implementation of interface conditions in a single drop flow problem we present a mesh moving method within a finite element context where the interface conditions of a two phase flow problem are conveniently included in suitable chosen subspaces of the general trial and testfunction spaces a weak formulation of the two phase flow problem including species transport is derived where the problem specific function spaces are replaced by appropriate projections operating on standard function spaces the transfer of the variational formulation to a finite element method is straightforward and results in one set of equations for both fluidic phases this subspace projection method is applicable to a wide range of interfacial conditions for multiphase flow problems the method is validated for single drop flow problems including species transfer furthermore an application to the simulation of stagnant caps is presented

eliminating spurious velocities with a stable approximation of viscous incompressible two phase stokes flow abstract we present a parametric finite element approximation of two phase flow this free boundary problem is given by the stokes equations in the two phases which are coupled via jump conditions across the interface using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface in addition the mesh quality of the parametric approximation of the interface does not deteriorate in general over time and an equidistribution property can be shown for a semidiscrete continuous in time variant of our scheme in two space dimensions moreover on using a simple xfem pressure space enrichment we obtain exact volume conservation for the two phase regions furthermore our fully discrete finite element approximation can be shown to be unconditionally stable we demonstrate the applicability of our method with some numerical results which in particular demonstrate that spurious velocities can be avoided in the classical test cases