poisson nernst planck equations for simulating biomolecular diffusion reaction processes i finite element solutions in this paper we developed accurate finite element methods for solving 3 d poisson nernst planck pnp equations with singular permanent charges for simulating electrodiffusion in solvated biomolecular systems the electrostatic poisson equation was defined in the biomolecules and in the solvent while the nernst planck equation was defined only in the solvent we applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules and formulated regular well posed pnp equations an inexact newton method was used to solve the coupled nonlinear elliptic equations for the steady problems while an adams bashforth crank nicolson method was devised for time integration for the unsteady electrodiffusion we numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the nernst planck equation and theoretically proved that the transformed formulation is always associated with an ill conditioned stiffness matrix we also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution the numerical methods are shown to be accurate and stable by various test problems and are applicable to real large scale biophysical electrodiffusion problems

where does the norm of a weighted polynomial live a characterization is given of the sets supporting the uniform norms of weighted polynomials w x npn x wherepn is any polynomial of degree at mostn the closed support ofw x may be bounded or unbounded of special interest is the case whenw x has a nonempty zero setz the treatment of weighted polynomials consists of associating each admissible weight with a certain functional defined on subsets of z one main result of this paper states that there is a unique compact set independent ofn andpn maximizing this functional that contains the points where the norms of weighted polynomials are attained the distribution of the zeros of chebyshev polynomials corresponding to the weights w x n is also studied the main theorems give a unified method of investigating many particular examples applications to weighted approximation on the real line with respect to a fixed weight are included

a parallelized meshfree method with boundary enrichment for large scale cfd a parallel computational implementation of a meshfree method the reproducing kernel particle method rkpm is used for 3 d implicit cfd analysis a novel procedure for implementing the essential boundary condition using the hierarchical enrichment method is presented using this enrichment along the essential boundaries produces results that more closely match experimental and analytical results for a flow past a cylinder problem than does either the finite element method or other meshfree methods that require matrix inversion for the application of essential boundary conditions this technique also allows the efficient parallelization of the algorithm and leads to higher parallel speedups than do other boundary condition implementations many of which involve inherently serial steps this is important because the expense of meshfree computations makes parallelization crucial for large size problems the performance of the parallelization technique and the accuracy of the implicit cfd algorithm are demonstrated in two example problems

error analysis of finite element method for poisson nernst planck equations in this paper we study the a priori error estimates of finite element method for the system of time dependent poisson nernst planck equations and for the first time we obtain its optimal error estimates in l h 1 and l 2 h 1 norms and suboptimal error estimates in l l 2 norm with linear element and optimal error estimates in l l 2 norm with quadratic or higher order element for both semi and fully discrete finite element approximations numerical experiments are also given to validate the theoretical results

extended immersed boundary method using fem and rkpm abstract the proposed extended immersed boundary method eibm has several distinct features in comparison with the immersed boundary ib method in the ib method the interaction between the immersed elastic boundary and the surrounding viscous fluid is replaced with an equivalent body force distribution within the fluid domain the key ingredient of the ib method is the enforcement of the energy input from the equivalent body force into the fluid domain to be the same at all time as that of the elastic forces within the ib in eibm instead of the volumeless immersed elastic boundary we consider the submerged elastic solid which occupies a finite volume within the fluid domain in particular we replace the kinematic and dynamic matching of the fluid solid interface and the effect of the submerged solid with a judiciously chosen collection of equivalent nodal forces calculated in the context of finite element formulations the employment of the finite element representation enables a realistic stress analysis for the submerged solid subject to large deformations finally in eibm we also replace the discretized delta function commonly used in the ib method with the kernel functions in meshless reproducing kernel particle method rkpm with this improvement using the same finite support domain we could improve the discretized delta function from c 1 to c n continuity where n is chosen according to the required smoothness and resolution of such functions and more importantly enables the use of non uniform meshing in the fluid domain these extensions may eventually open doors to the modelling of complex biological fluid involving molecular cellular and flexible vessel flow interactions

current trends in asymptotics some problems and some solutions abstract i discuss five topics of current interest in asymptotic analysis the use of probabilistic methods to estimate the growth of combinatorial sequences asymptotic methods in the theory of random walks the asymptotic estimation of solutions of difference equations asymptotic expansions in generalized scales and the computation by asymptotic methods of distributions whose moments are known i give a number of examples and mention some outstanding unsolved problems

exact multiplicity of positive solutions for a class of semilinear problem ii we consider the positive solutions to the semilinear problem u f u 0 in b n u 0 on b n where bn is the unit ball in rn n 1 and is a positive parameter it is well known that if is a smooth function then any positive solution to the equation is radially symmetric and all solutions can be parameterized by their maximum values we develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions u and the precise shape of the global bifurcation diagrams are rigorously proved our technique combines the bifurcation analysis stability analysis and topological methods we show that the shape of the bifurcation curve depends on the shape of the graph of function u u as well as the growth rale of

a bubble stabilized finite element method for dirichlet constraints on embedded interfaces we examine a bubble stabilized finite element method for enforcing dirichlet constraints on embedded interfaces by embedded we refer to problems of general interest wherein the geometry of the interface is assumed independent of some underlying bulk mesh as such the robust imposition of dirichlet constraints using a lagrange multiplier field is not trivial to focus issues we consider a simple one sided problem that is representative of a wide class of evolving interface problems the bulk field is decomposed into coarse and fine scales giving rise to coarse scale and fine scale one sided sub problems the fine scale solution is approximated with bubble functions permitting static condensation and giving rise to a stabilized form bearing strong analogy with a classical method importantly the method is simple to implement readily extends to multiple dimensions obviates the need to specify any free stabilization parameters and can lead to a symmetric positive definite system of equations the performance of the method is then examined through several numerical examples the accuracy of the lagrange multiplier is compared to results obtained using a local version of the domain integral method the variational multiscale approach proposed herein is shown to both stabilize the lagrange multiplier and improve the accuracy of the post processed fluxes copyright 2006 john wiley sons ltd

the steady boundary value problem for charged incompressible fluids pnp navier stokes systems abstract the initial boundary value problem for the poisson nernst planck navier stokes model was investigated in j w jerome r sacco global weak solutions for an incompressible charged fluid with multi scale couplings initial boundary value problem nonlinear anal 71 2009 e2487 e2497 where an existence theory was demonstrated based upon rothe s method of horizontal lines in this article the steady case is considered and the existence of a weak solution is established for the boundary value problem this solution satisfies a weak maximum principle for the concentrations relative to the boundary values as noted in the above mentioned citation the model assumes significance because of its connection to the electrophysiology of the cell including neuronal cell monitoring and microfluidic devices in biochip technology the model has also been used in other applications including electro osmosis the steady model is especially important in ion channel modeling because the channel remains open for milliseconds and the transients appear to decay on the scale of tens of nanoseconds

the set of positive solutions of semilinear equations in large balls the exact number of positive solutions of u f u 0 on finite balls in n is determined the assumptions about f u are similar to those imposed by serrin and the second author in a previous study of uniqueness of the positive solution when the spatial domain is all of n see 7 8 for finite balls of sufficiently large radius it is shown here that there are exactly two positive and hence radial solutions to this end we first prove the linear nondegeneracy of the positive solution of n this is obtained by applying the technique of monotone separation of graphs 7 to the linearised equations somewhat sharper estimates are required here see part i section 2

immersed finite element method abstract in this paper the immersed finite element method ifem is proposed for the solution of complex fluid and deformable structure interaction problems encountered in many physical models in ifem a lagrangian solid mesh moves on top of a background eulerian fluid mesh which spans over the entire computational domain hence the mesh generation is greatly simplified moreover both fluid and solid domains are modeled with the finite element methods and the continuity between the fluid and solid sub domains are enforced via the interpolation of the velocities and the distribution of the forces with the reproducing kernel particle method rkpm delta function in comparison with the immersed boundary ib method the higher ordered rkpm delta function enables the fluid domain to have nonuniform spatial meshes with arbitrary geometries and boundary conditions the use of such kernel functions may eventually open doors to multi scale and multi resolution modelings of complex fluid structure interaction problems rigid and deformable spheres dropping in channels are simulated to demonstrate the unique capabilities of the proposed method the results compare well with the experimental data to the authors knowledge these are the first solutions that deal with particulate flows with very flexible solids

convergent discretizations for the nernst planck poisson system we propose and compare two classes of convergent finite element based approximations of the nonstationary nernst planck poisson equations whose constructions are motivated from energy versus entropy decay properties for the limiting system solutions of both schemes converge to weak solutions of the limiting problem for discretization parameters tending to zero our main focus is to study qualitative properties for the different approaches at finite discretization scales like conservation of mass non negativity discrete maximum principle decay of discrete energies and entropies to study long time asymptotics

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

a mixed finite element discretization of the energy transport model for semiconductors energy transport models describe the flow of electrons through a semiconductor device influenced by diffusive electrical and thermal effects they consist of the continuity equations for the mass and energy coupled with poisson s equation for the electrostatic potential the energy transport model can be written in a drift diffusion formulation which is used for the numerical approximation the stationary equations are discretized with an exponential fitting mixed finite element method in two space dimensions numerical simulations of a ballistic diode are performed and numerical convergence rates are computed furthermore a two dimensional metal semiconductor field effect transistor device with parabolic band structure is simulated

exact multiplicity of positive solutions for a class of semilinear problems we establish an exact multiplicity result of positive solutions of u f u 0inbn u 0on bn wherebnis the unit ball fsatisfiesf changes sign only once and asymptotically sublinear or linear the nonlinearities which we are concerned with here includef u u u b c u for 0 2b candf u up uqfor 1 p q precise global bifurcation diagrams are obtained

rigorous homogenization of a stokes nernst planck poisson system abstract we perform the periodic homogenization i e e 0 of a non stationary stokes nernst planck poisson system using two scale convergence where e is a suitable scale parameter the objective is to investigate the influence of different boundary conditions and variable choices of scalings in e of the microscopic system of partial differential equations on the structure of the upscaled limit model equations due to the specific nonlinear coupling of the underlying equations special attention has to be paid when passing to the limit in the electrostatic drift term as a direct result of the homogenization procedure various classes of upscaled model equations are obtained

mixed finite element analysis for the poisson nernst planck stokes coupling abstract in this paper a type of mixed finite element method is developed to solve the poisson nernst planck stokes coupling problem which is adopted to model charged fluids through the transport coupling between stokes equations of an incompressible fluid and poisson nernst planck pnp equations of a diffuse charge system the taylor hood p k 1 p k mixed element is employed to discretize both mixed poisson equations and stokes equations and the standard p k finite element is used to discretize nernst planck equations optimal convergence rates for both the electrostatic potential and ionic concentrations of pnp equations are obtained in both l 2 and h 1 norms simultaneously optimal convergence rates are also obtained for the velocity and pressure of stokes equations in h 1 d and l 2 norm respectively numerical experiments validate the theoretical results

immersed boundary methods for general finite difference and finite volume navier stokes solvers we present an immersed boundary algorithm for incompressible flows with complex boundaries suitable for cartesian or curvilinear grid system the key stages of any immersed boundary technique are the interpolation of a velocity field given on a mesh onto a general boundary a line in 2d a surface in 3d and the spreading of a force field from the immersed boundary to the neighboring mesh points to enforce the desired boundary conditions on the immersed boundary points we propose a technique that uses the reproducing kernel particle method w k liu s jun y f zhang reproducing kernel particle methods int j numer methods fluids 20 8 1995 1081 1106 for the interpolation and spreading unlike other methods presented in the literature the one proposed here has the property that the integrals of the force field and of its moment on the grid are conserved independent of the grid topology uniform or non uniform cartesian or curvilinear the technique is easy to implement and is able to maintain the order of the original underlying spatial discretization applications to two and three dimensional flows in cartesian and non cartesian grid system with uniform and non uniform meshes are presented

consistency of semiconductor modeling an existence stability analysis for the stationary van roosbroeck system this paper examines the solution map of the stationary version of the van roosbroeck model for the flow of electrons and holes in a crystalline semiconductor thermodynamic equilibrium and the einstein relations linking mobility and diffusion are assumed to hold for this model incorporated into the model is the quantum statistical assumption that the carrier densities satisfy fermi dirac statistical laws with localization in the boltzmann tail this latter approximation which is seen to be inessential characterizes the conduction and valence bands as nondegenerate the boundary conditions are of two types and correspond to applied potential differences and doping on the contact portions of the device and insulation on the remainder the complete system then involves a coupled set of three nonlinear partial differential equations with their boundary conditions for the electrostatic potential and for the quasi fermi levels corresponding to the electron and hole concentrations in the absence of the classical nondegeneracy approximation the model remains valid but without a natural physical interpretation for two ofthe three dependent variables we demonstrate the existence ofsolutions for this system by associating a solution mapping t such that the fixed points of t define solutions of the system a priori estimates or weak maximum principles permit t to act invariantly on a naturally chosen subset of a hilbert product space the definition of t roughly follows the gummel procedure for uncoupling systems the schauder fixed point theorem is used to prove the existence of a fixed point for t uniqueness is not demonstrated and very possibly does not hold for these systems without special restrictions on the boundary conditions we also introduce a discrete version of the mapping t which is defined in terms of a family of finite difference approximations this mapping enjoys the stability property that it acts invariantly on a rectangular region in euclidean space the dimensions are the same as those specified in the pointwise bounds which define the domain of t a second existence theory framework with quasi constructive aspects is also studied this is based upon pseudomonotone operators the presentation is technology independent except for a section on mos fet devices these typically induce a breakdown ofuniform ellipticity and may be handled by modifications of the major arguments they also exemplify a class of devices for which the electrostatic field is defined on a strictly larger domain than the carriers themselves 1 introduction and model description this paper provides a mathematical analysis of a stationary system solution map which forms the basis for the highly successful computational algorithms used for the numerical solution of the semiconductor system cf bank rose and fichtner 3 our methods involve pointwise stability estimates on solution components organized in such a way that the solution map and a family of discrete approximations may be studied in parallel if desired moreover the weak solution methods do not require regularity theory hence are applicable to physical domains commonly encountered in practice such domains may include re entrant corner points etc and typically induce degenerate elliptic systems we shall also develop the model from physical principles the reason for this is not primarily to compile source material from several different references but rather to indicate clearly to the reader what the level of generality is and to facilitate the discussion of important variations in the device models the models studied herein are reaction diffusion con vection models for which the einstein relations linking carrier mobility and diffusion are assumed to hold the dependent variables of the system include one potential variable inducing an ambient field and two concentration variables for carriers whose convective movement is field dependent this suggests a fairly rich set ofpossible

an elliptic singularly perturbed problem with two parameters ii robust finite element solution in this paper we consider a singularly perturbed elliptic model problem with two small parameters posed on the unit square the problem is solved numerically by the finite element method using piecewise linear or bilinear elements on a layer adapted shishkin mesh we prove that method with bilinear elements is uniformly convergent in an energy norm numerical results confirm our theoretical analysis

a hybrid variational collocation immersed method for fluid structure interaction using unstructured t splines summary we present a hybrid variational collocation immersed and fully implicit formulation for fluid structure interaction fsi using unstructured t splines in our immersed methodology we define an eulerian mesh on the whole computational domain and a lagrangian mesh on the solid domain which moves arbitrarily on top of the eulerian mesh mathematically the problem reduces to solving three equations namely the linear momentum balance mass conservation and a condition of kinematic compatibility between the lagrangian displacement and the eulerian velocity we use a weighted residual approach for the linear momentum and mass conservation equations but we discretize directly the strong form of the kinematic relation deriving a hybrid variational collocation method we use t splines for both the spatial discretization and the information transfer between the eulerian mesh and the lagrangian mesh t splines offer us two main advantages against non uniform rational b splines they can be locally refined and they are unstructured the generalized method is used for the time discretization we validate our formulation with a common fsi benchmark problem achieving excellent agreement with the theoretical solution an example involving a partially immersed solid is also solved the numerical examples show how the use of t junctions and extraordinary nodes results in an accurate efficient and flexible method copyright 2015 john wiley sons ltd

a finite element approximation theory for the drift diffusion semiconductor model two sided estimates are derived for the approximation of solutions to the drift diffusion steady state semiconductor device system which are identified with fixed points of gummel s solution map the approximations are defined in terms of fixed points of numerical finite element discretization maps by use of a calculus developed by krasnosel skii and his coworkers it is possible both to locate approximations near fixed points in an a priori manner as well as fixed points near approximations in an a posteriors manner these results thus establish a nonlinear approximation theory in the energy norm with rate keyed to what is possible in a standard linear theory this analysis provides a convergence theory for typical computational approaches in current use for semiconductor simulation

numerical methods on shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy abstract a numerical study is made for a class of boundary value problems of second order differential equations in which the highest order derivative is multiplied by a small parameter and both the differentiated convection and undifferentiated reaction terms are with negative shift we analyze three difference operators l k h k 1 2 3 a simple upwind scheme midpoint upwind scheme and a hybrid scheme respectively on a shishkin mesh to approximate the solution of the problem theoretical error bounds are established the accuracy by one of the grid adaptation strategies namely grid redistribution is examined by solving the considered problem using hybrid method for some values of and n as a result the accuracy gets improved by this grid adaptation strategy with almost the same computational cost a few numerical results exhibiting the performance of these three schemes are presented

well posedness for the navier stokes nernst planck poisson system in triebel lizorkin space and besov space with negative indices abstract this paper is concerned with the well posedness of the navier stokes nerst planck poisson system nsnpp let s p 2 n p we prove that the nsnpp has a unique local solution u v w e u t e v t e v t for u 0 v 0 w 0 in a subspace i e v u 1 v v 1 v v 1 of f 1 2 b p s p b p s p with u 0 0 we also prove that there exists a unique small global solution u v w e u e v e v for any small initial data u 0 v 0 w 0 f 1 2 b p s p b p s p with u 0 0