some upwinding techniques for finite element approximations of convection diffusion equations a uniform framework for the study of upwinding schemes is developed the standard finite element galerkin discretization is chosen as the reference discretization and differences between other discretization schemes and the reference are written as artificial diffusion terms these artificial diffusion terms are spanned by a four dimensional space of element diffusion matrices three basis matrices are symmetric rank one diffusion operators associated with the edges of the triangle the fourth basis matrix is skew symmetric and is associated with a rotation by 2 while finite volume discretizations may be written as upwinded galerkin methods the converse does not appear to be true our approach is used to examine several upwinding schemes including the streamline diffusion method the box method the scharfetter gummel discretization and a divergence free scheme

the discrete maximum principle for linear simplicial finite element approximations of a reaction diffusion problem abstract this paper provides a sufficient condition for the discrete maximum principle for a fully discrete linear simplicial finite element discretization of a reaction diffusion problem to hold it explicitly bounds the dihedral angles and heights of simplices in the finite element partition in terms of the magnitude of the reaction coefficient and the spatial dimension as a result it can be computed how small the acute simplices should be for the discrete maximum principle to be valid numerical experiments suggest that the bound which considerably improves a similar bound in p g ciarlet p a raviart maximum principle and uniform convergence for the finite element method comput methods appl mech eng 2 1973 17 31 is in fact sharp

some discrete maximum principles arising for nonlinear elliptic finite element problems the discrete maximum principle dmp is an important measure of the qualitative reliability of the applied numerical scheme for elliptic problems this paper starts with formulating simple sufficient conditions for the matrix case and for nonlinear forms in banach spaces then a dmp is derived for finite element solutions for certain nonlinear partial differential equations we address nonlinear elliptic problems with mixed boundary conditions and interface conditions allowing possibly degenerate nonlinearities and thus extending our previous results

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

numerical schemes for hamilton jacobi equations on unstructured meshes in this paper a numerical scheme is presented by applying the finite element method to the viscosity equations of the hamilton jacobi equations on unstructured meshes by improving the finite element scheme another numerical scheme is constructed under certain limitations the numerical solutions of the two schemes converge to the viscosity solutions of the hamilton jacobi equations the latter numerical scheme has weaker restrictions than the former scheme for convergence numerical examples are provided to test the stability convergence and sensitivity to different meshes

a spectral analysis of the decoupling algorithm for semiconductor simulation the location in the complex plane of the eigenvalues of the derivatives of the partial nonlinear maps which constitute the fixed point mapping t defining gummel s method for decoupling the steady state semiconductor equations is examined for the general n dimensional case in particular it is proven that the size of these eigenvalues is unconditionally smaller than 1 it is shown that for up to three dimensional models all the derivatives are suitably compact this implies that the spectral radii of these mappings are equal to the size of the largest eigenvalue for the current continuity equations this implies an essential improvement over earlier contraction mapping results furthermore the precise location of the eigenvalues of the derivatives of the maps defined through the current continuity equations for the general one dimensional case is found for a special one dimensional case it is proven that the size of the eigenvalues of the total mapping t defining gummel s method is unconditionally smal

on the stable solution of transient convection diffusion equations a transient convection diffusion equation is considered which particularly arises in optimization problems for the asymmetric flow field flow fractionation af4 process a time dependent generalization of the monotone edge averaged finite element eafe scheme is used to obtain a stable discretization in the convection dominated regime new error estimates are proved for this scheme and a comparison with the popular supg method is presented numerical experiments demonstrate that the eafe method is more suitable for problems where boundary layers are formed we consider transient convection diffusion equation with dominating convection the model has an application in optimal control of the asymmetric flow field flow fractionation process for a stable discretization we propose a monotone edge averaged finite element eafe scheme eafe is generalized to a time dependent case and a new error estimate is proved we present numerical results for comparison with a popular supg method

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

multigrid methods for convection diffusion problems discretized by a monotone scheme abstract we study multigrid mg methods for the solution of systems of linear algebraic equations obtained from a stable discretization of convection diffusion problems by an exponential fitting scheme the latter ensures the stability of the simplest possible coarse grid operators obtained from galerkin projections based on graph matching linear and nonlinear mg preconditioners are defined in the framework of algebraic multilevel iteration the option of using polynomial smoothers is investigated in context of nonsymmetric problems and a systematic performance comparison is presented for various algorithms on a representative set of two and three dimensional test problems

spatial random permutations and infinite cycles we consider systems of spatial random permutations where permutations are weighed according to the point locations infinite cycles are present at high densities the critical density is given by an exact expression we discuss the relation between the model of spatial permutations and the ideal and interacting quantum bose gas

stabilized galerkin approximation of convection diffusion reaction equations discrete maximum principle and convergence we analyze a nonlinear shock capturing scheme for h1 conforming piecewise affine finite element approximations of linear elliptic problems the meshes are assumed to satisfy two standard conditions a local quasi uniformity property and the xu zikatanov condition ensuring that the stiffness matrix associated with the poisson equation is an m matrix a discrete maximum principle is rigorously established in any space dimension for convection diffusion reaction problems we prove that the shock capturing finite element solution converges to that without shock capturing if the cell peclet numbers are sufficiently small moreover in the diffusion dominated regime the difference between the two finite element solutions super converges with respect to the actual approximation error numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates

efficiency and proper efficiency in nonlinear vector maximum problems abstract a nonlinear vector maximum problem nlvmp is considered an equivalent vector maximum problem vmp is formulated by linearizing the objectives of nlvmp assuming generalized convexity and a constraint qualification necessary and sufficient conditions in both kuhn tucker and isermann form are obtained for an efficient solution of nlvmp also if objectives are pseudolinear and satisfy a certain boundedness condition it is proved that every efficient solution of nlvmp is properly efficient finally a numerical example is discussed to illustrate the results

on some mixed finite element methods with numerical integration in this paper a new family of mixed finite volume methods is analyzed for the approximation of a reaction diffusion problem all the methods are obtained starting from the dual mixed formulation of the problem and then employing the lowest order raviart thomas finite element spaces plus a suitable quadrature formula for the mass matrix this allows for the use of different averages of the inverse diffusion coefficient to enforce the constitutive law for the fluxes at the interelement boundary in a finite volume fashion the soundness of the methods is supported by an error analysis which shows optimal cal o h convergence rate with respect to the standard mixed finite element norm in the case of both smooth and piecewise smooth coefficients numerical results on test problems with both smooth and nonsmooth coefficients support the theoretical estimates

loop correlations in random wire models we introduce a family of loop soup models on the hypercubic lattice the models involve links on the edges and random pairings of the link endpoints on the sites we conjecture that loop correlations of distant points are given by poisson dirichlet correlations in dimensions three and higher we prove that in a specific random wire model that is related to the classical xy spin system the probability that distant sites form an even partition is given by the poisson dirichlet counterpart

on the a posteriori error analysis for linear fokker planck models in convection dominated diffusion problems abstract this work is aimed at the derivation of reliable and efficient a posteriori error estimates for convection dominated diffusion problems motivated by a linear fokker planck problem appearing in computational neuroscience we obtain computable error bounds of functional type for the static and time dependent case and for different boundary conditions mixed and pure neumann boundary conditions finally we present a set of various numerical examples including discussions on mesh adaptivity and space time discretisation the numerical results confirm the reliability and efficiency of the error estimates derived

failure of the discrete maximum principle for an elliptic finite element problem there has been a long standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem this is related to knowing whether the discrete green s function is positive for triangular meshes allowing sufficiently good approximation of h 1 functions we study this question for the poisson problem in two dimensions discretized via the galerkin method with continuous piecewise linears we give examples which show that in general the answer is negative and furthermore we extend the number of cases where it is known to be positive our techniques utilize some new results about discrete green s functions that are of independent interest

hierarchical electrochemical modeling and simulation of bio hybrid interfaces abstract in this article we propose and investigate a hierarchy of mathematical models based on partial differential equations pde and ordinary differential equations ode for the simulation of the biophysical phenomena occurring in the electrolyte fluid that connects a biological component a single cell or a system of cells and a solid state device a single silicon transistor or an array of transistors the three members of the hierarchy ordered by decreasing complexity are i a 3d poisson nernst planck pnp pde system for ion concentrations and electric potential ii a 2d reduced pnp system for the same dependent variables as in i iii a 2d area contact pde system for electric potential coupled with a system of odes for ion concentrations the backward euler method is adopted for temporal semi discretization and a fixed point iteration based on gummel s map is used to decouple system equations spatial discretization is performed using piecewise linear triangular finite elements stabilized via edge based exponential fitting extensively conducted simulation results are in excellent agreement with existing analytical solutions of the pnp problem in radial coordinates and experimental and simulated data using simplified lumped parameter models

transport of charged particles entropy production and maximum dissipation principle abstract in order to describe the dynamics of crowded ions charged particles we use an energetic variational approach to derive a modified poisson nernst planck pnp system which includes an extra dissipation due to the effective velocity differences between ion species such a system has more complicated nonlinearities than the original pnp system but with the same equilibrium states using galerkin s method and schauder s fixed point theorem we develop a local existence theorem of classical solutions for the modified pnp system different dynamics but same equilibrium states between the original and modified pnp systems can be represented by numerical simulations using finite element method techniques

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

on nonobtuse simplicial partitions this paper surveys some results on acute and nonobtuse simplices and associated spatial partitions these partitions are relevant in numerical mathematics including piecewise polynomial approximation theory and the finite element method special attention is paid to a basic type of nonobtuse simplices called path simplices the generalization of right triangles to higher dimensions in addition to applications in numerical mathematics we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics

conditioning of the steady state semiconductor device problem when solving numerically the steady state semiconductor device problem using appropriate discretizations extremely large condition numbers are often encountered for the linearized discrete device problem these condition numbers are so large that if they represented a sharp bound on the amplification of input errors or even of roundoff errors then the obtained numerical solution would be meaningless as it turns out one major reason for these large numbers is due to poor row and column scaling which is essentially harmless and or can be fixed but another reason could be an ill conditioned device which yields a true loss of significant digits in the numerical calculation in this paper we carry out a conditioning analysis for the steady state device problem we consider various quasilinearizations as well as gummel type iterations and obtain stability bounds which may indeed allow ill conditioning in general these bounds are exponential in the potential variation and are sharp e g for a thyristor but for devices where each smooth subdomain has an ohmic contact e g a pn diode moderate bounds guaranteeing well conditioning are obtained moreover the analysis suggests how various row and column scalings should be applied in order for the measured condition numbers to correspond more realistically to the true loss of significant digits in the calculations

blending low order stabilised finite element methods a positivity preserving local projection method for the convection diffusion equation in this work we propose a nonlinear blending of two low order stabilisation mechanisms for the convection diffusion equation the motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions the approach is to blend a first order artificial diffusion method which will be active only in the vicinity of layers and extrema with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution we prove existence of discrete solutions as well as convergence under appropriate assumptions on the nonlinear terms and on the exact solution numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle while the layers are not smeared significantly

optimality theorems for convex semidefinite vector optimization problems abstract in this paper we consider a convex semidefinite vector optimization problem sdvp involving a convex objective vector function a matrix linear inequality constraint and a geometric constraint and define properly weakly efficient solutions for sdvp as we do for ordinary vector optimization problems we present necessary and sufficient optimality conditions for efficient solutions of sdvp which hold without any constraint qualification also we present necessary and sufficient optimality conditions for properly efficient solutions and weakly efficient solutions for sdvp which hold without any constraint qualification furthermore we give a sufficient condition for an efficient solution of sdvp to be properly efficient

an analysis of a conforming exponentially fitted finite element method for a convection diffusion problem in this paper we present a convergence analysis for a conforming exponentially fitted galerkin finite element method with triangular elements for a linear singularly perturbed convection diffusion problem with a singular perturbation parameter e it is shown that the error for the finite element solution in the energy norm is bounded by o h e1 2 u 2 e 1 2 u 1 if a regular family of triangular meshes is used in the case that a problem contains only exponential boundary layers the method is shown to be convergent at a rate of h1 2 h ln e on anisotropic layer fitted meshes

a discretization scheme for an extended drift diffusion model including trap assisted phenomena an extended drift diffusion model is considered to account for the kinetics of electrons trapped in defect states within a semiconductor material a discretization scheme based on newton krylov iterations and mixed finite volumes is then proposed and applied to the model even in the presence of schottky contacts i e robin type boundary conditions numerical results concerning the simulation of an electro optical device in several working conditions are presented last

nonnegativity of exact and numerical solutions of some chemotactic models we investigate nonnegativity of exact and numerical solutions to a generalized keller segel model this model includes the so called minimal keller segel model but can cover more general chemistry we use maximum principles and invariant sets to prove that all components of the solution of the generalized model are nonnegative we then derive numerical methods using finite element techniques for the generalized keller segel model adapting the ideas in our proof of nonnegativity of exact solutions to the discrete setting we are able to show nonnegativity of discrete solutions from the numerical methods under certain standard assumptions one of the numerical methods is then applied to the minimal keller segel model recalling known results on the qualitative behavior of this model we are able to choose parameters that yield convergence to a nonhomogeneous stationary solution while proceeding to exhibit these stationary patterns we also demonstrate how naive choices of numerical methods can give physically unrealistic solutions thereby justifying the need to study positivity preserving methods

variational space time elements for large scale systems abstract in this paper we introduce a new galerkin based formulation for transient continuum problems governed by partial differential equations in space and time therefore we aim at a direct finite element discretization of the space time suitable for massive parallel analysis of the arising large scale problem the proposed formulation is applied to thermal mechanical and fluid systems as well as to a kuramoto sivashinsky problem representing the general class of higher order formulations in material science using nurbs based shape functions we verify whenever possible the conservation properties of the formulation finally a series of examples demonstrate the applicability to all systems presented in this paper

asymptotic statistics of cycles in surrogate spatial permutations we propose an extension of the ewens measure on permutations by choosing the cycle weights to be asymptotically proportional to the degree of the symmetric group this model is primarily motivated by a natural approximation to the so called spatial random permutations recently studied by betz and ueltschi hence the name surrogate spatial but it is of substantial interest in its own right we show that under the suitable thermodynamic limit both measures have the similar critical behaviour of the cycle statistics characterized by the emergence of infinitely long cycles moreover using a greater analytic tractability of the surrogate spatial model we obtain a number of new results about the asymptotic distribution of the cycle lengths both small and large in the full range of subcritical critical and supercritical domains in particular in the supercritical regime there is a parametric phase transition from the poisson dirichlet limiting distribution of ordered cycles to the occurrence of a single giant cycle our techniques are based on the asymptotic analysis of the corresponding generating functions using polya s enumeration theorem and complex variable methods

pseudolinearity and efficiency first order and second order characterizations of pseudolinear functions are derived for a nonlinear programming problem involving pseudolinear functions only it is proved that every efficient solution is properly efficient under some mild conditions

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

inverse average type finite element discretizations of selfadjoint second order elliptic problems etude d une classe de discretisations a elements finis lineaires par morceaux des problemes aux valeurs limites elliptiques du second ordre autoadjoints

nonstationary monotone iterative methods for nonlinear partial differential equations a simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations with the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system picard gauss seidel and jacobi monotone iterative methods are presented and analyzed for the adaptive solutions the adaptive meshes are generated by the 1 irregular mesh refinement scheme which together with the m matrix of the finite element stiffness matrix lead to existence uniqueness comparison theorems with simple upper and lower solutions as initial iterates some numerical examples including a test problem with known analytical solution are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties numerical results of simulations on a mosfet with the gate length down to 34 nm are also given

limit distributions for euclidean random permutations we study the length of cycles in the model of spatial random permutations in euclidean space in this model for given length l density rho dimension d and jump density varphi one samples rho l d particles in a d dimensional torus of side length l and a permutation pi of the particles with probability density proportional to the product of values of varphi at the differences between a particle and its image under pi the distribution may be further weighted by a factor of theta to the number of cycles in pi following matsubara and feynman the emergence of macroscopic cycles in pi at high density rho has been related to the phenomenon of bose einstein condensation for each dimension d ge 1 we identify sub critical critical and super critical regimes for rho and find the limiting distribution of cycle lengths in these regimes the results extend the work of betz and ueltschi our main technical tools are saddle point and singularity analysis of suitable generating functions following the analysis by bogachev and zeindler of a related surrogate spatial model

current voltage characteristics simulation of semiconductor devices using domain decomposition abstract we study the current voltage characteristics of one dimensional semiconductor devices by numerical approximation based on finite elements of the steady state semiconductor device equations a block nonlinear gauss seidel procedure is employed to decouple the full system then at each iteration a neumann neumann domain decomposition method is applied to solve the linearized equations numerical examples will be given with special emphasis on charge generation effects due to impact ionization

edge based nonlinear diffusion for finite element approximations of convection diffusion equations and its relation to algebraic flux correction schemes for the case of approximation of convection diffusion equations using piecewise affine continuous finite elements a new edge based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle the diffusion operator is shown to be lipschitz continuous and linearity preserving using these properties we provide a full stability and error analysis which in the diffusion dominated regime shows existence uniqueness and optimal convergence then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters the performance of the method is illustrated on some numerical test cases in two space dimensions

a second order overlapping schwarz method for a 2d singularly perturbed semilinear reaction diffusion problem an overlapping schwarz domain decomposition is applied to a semilinear reaction diffusion equation posed in a smooth two dimensional domain the problem may exhibit multiple solutions its diffusion parameter e2 is arbitrarily small which induces boundary layers the schwarz method invokes a boundary layer subdomain and an interior subdomain the narrow subdomain overlap being of width o e ln h where h is the maximum side length of mesh elements and the global number of mesh nodes does not exceed o h 2 we employ finite differences on layer adapted meshes of bakhvalov and shishkin types in the boundary layer subdomain and lumped mass linear finite elements on a quasiuniform delaunay triangulation in the interior subdomain for this iterative method we present maximum norm error estimates for e 0 1 it is shown in particular that when e c ln h 1 one iteration is sufficient to get second order convergence with in the case of the shishkin mesh a logarithmic factor in the maximum norm uniformly in e numerical results are presented to support our theoretical conclusions

a proof of convergence of gummel s algorithm for realistic device geometries convergence is proven for a typical example from the class of highly successful decoupling algorithms for semiconductor simulation collectively known as gummel s method for one two and three dimensional models because a nonlinear equation is solved for the potential u at every step the considered version corresponds closely to the algorithms used for numerical computation in practice as opposed to most earlier publications the dependence of the regularity of the solution on the device geometry and the nature of the boundary conditions for the system of mixed boundary value problems is considered by a detailed analysis of the boundary conditions for a typical two or three dimensional device a theoretical regularity result by murthy and stampacchia can be applied hence it follows that for a physically realistic device geometry the solution is sufficiently regular for the algorithm to converge

a conservative and monotone mixed hybridized finite element approximation of transport problems in heterogeneous domains in this article we discuss the numerical approximation of transport phenomena occurring at material interfaces between physical subdomains with heterogenous properties the model in each subdomain consists of a partial differential equation with diffusive convective and reactive terms the coupling between each subdomain being realized through an interface transmission condition of robin type the numerical approximation of the problem in the two dimensional case is carried out through a dual mixed hybridized finite element method with numerical quadrature of the mass flux matrix the resulting method is a conservative finite volume scheme over triangular grids for which a discrete maximum principle is proved under the assumption that the mesh is of delaunay type in the interior of the domain and of weakly acute type along the domain external boundary and internal interface the stability accuracy and robustness of the proposed method are validated on several numerical examples motivated by applications in biology electrophysiology and neuroelectronics

numerical homogenization method for parabolic advection diffusion multiscale problems with large compressible flows we introduce a numerical homogenization method based on a discontinuous galerkin finite element heterogeneous multiscale method to efficiently approximate the effective solution of parabolic advection diffusion problems with rapidly varying coefficients large peclet number and compressible flows to estimate the missing data of an effective model numerical upscaling is performed which accurately captures the effects of microscopic solenoidal or gradient flow at a macroscopic scale such as enhancement or depletion of the effective diffusion for compressible flow with periodic data we derive sharp a priori error estimates for the macro and micro discretization errors which are robust in the advection dominated regime numerical tests confirm the error estimates for problems with periodic data and illustrate the applicability of our method for problems with non periodic data

energetically stable discretizations for charge transport and electrokinetic models a finite element discretization using a method of lines approached is proposed for approximately solving the poisson nernst planck pnp equations this discretization scheme enforces positivity of the computed solutions corresponding to particle density functions and a discrete energy estimate is established that takes the same form as the energy law for the continuous pnp system this energy estimate is extended to finite element solutions to an electrokinetic model which couples the pnp system with the incompressible navier stokes equations numerical experiments are conducted to validate convergence of the computed solution and verify the discrete energy estimate

spatial random permutations with small cycle weights we consider the distribution of cycles in two models of random permutations that are related to one another in the first model cycles receive a weight that depends on their length the second model deals with permutations of points in the space and there is an additional weight that involves the length of permutation jumps we prove the occurrence of infinite macroscopic cycles above a certain critical density

discrete maximum principle for higher order finite elements in 1d we formulate a sufficient condition on the mesh under which we prove the discrete maximum principle dmp for the one dimensional poisson equation with dirichlet boundary conditions discretized by the hp fem the dmp holds if a relative length of every element k in the mesh is bounded by a value h rel p 0 9 1 where p 1 is the polynomial degree of the element k the values h rel p are calculated for 1 p 100

monotonicity preserving finite element schemes based on differentiable nonlinear stabilization in this work we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping the method relies on an artificial diffusion method based on a graph laplacian operator it is nonlinear since it depends on a shock detector further the resulting method is linearity preserving the same shock detector is used to gradually lump the mass matrix the resulting method is led positivity preserving and also satisfies a global dmp lipschitz continuity has also been proved however the resulting scheme is highly nonlinear leading to very poor nonlinear convergence rates we propose a smooth version of the scheme which leads to twice differentiable nonlinear stabilization schemes it allows one to straightforwardly use newton s method and obtain quadratic convergence in the numerical experiments steady and transient linear transport and transient burgers equation have been considered in 2d using the newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of anderson acceleration with the original non smooth scheme in any case these properties are only true for the converged solution but not for iterates in this sense we have also proposed the concept of projected nonlinear solvers where a projection step is performed at the end of every nonlinear iterations onto a fe space of admissible solutions the space of admissible solutions is the one that satisfies the desired monotonic properties maximum principle or positivity

maximum principle in linear finite element approximations of anisotropic diffusion convection reaction problems a mesh condition is developed for linear finite element approximations of anisotropic diffusion convection reaction problems to satisfy a discrete maximum principle loosely speaking the condition requires that the mesh be simplicial and mathcal o vert varvec b vert infty h vert c vert infty h 2 o b h c h 2 nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix where h h denotes the mesh size and varvec b b and c c are the coefficients of the convection and reaction terms in two dimensions the condition can be replaced by a weaker mesh condition an mathcal o vert varvec b vert infty h vert c vert infty h 2 o b h c h 2 perturbation of a generalized delaunay condition these results include many existing mesh conditions as special cases numerical results are presented to verify the theoretical findings

preserving nonnegativity of an affine finite element approximation for a convection diffusion reaction problem an affine finite element scheme approximation of a time dependent linear convection diffusion reaction problem in 2d and 3d is presented for these equations which do not satisfy an underlying maximum principle sufficient conditions are given in terms of the coefficient functions the computational grid and the discretization parameters to ensure that the nonnegativity property of the true solution is also satisfied by its approximation numerical examples are given which confirm the necessity and sufficiency of the discretization conditions to guarantee the nonnegativity of the approximation we present a monotone fem scheme for a convection diffusion reaction problem in 2 3d the considered equation does not possess an underlying maximum principle sufficient conditions are given to ensure the nonnegativity of the approximations numerical examples confirm the necessity and sufficiency of the conditions

domain decomposition for multiscale pdes we consider additive schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic pdes with highly variable coefficients in contrast to standard analyses we do not assume that the coefficients can be resolved by a coarse mesh this situation arises often in practice for example in the computation of flows in heterogeneous porous media in both the deterministic and monte carlo simulated stochastic cases we consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid we perform a new analysis of the preconditioned matrix which shows rather explicitly how its condition number depends on the variable coefficient in the pde as well as on the coarse mesh and overlap parameters the classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements by contrast our new results show that with a good choice of subdomains and coarse space basis functions the preconditioner can still be robust even for large coefficient variation inside domains when the classical method fails to be robust in particular our estimates prove very precisely the previously made empirical observation that the use of low energy coarse spaces can lead to robust preconditioners we go on to consider coarse spaces constructed from multiscale finite elements and prove that preconditioners using this type of coarsening lead to robust preconditioners for a variety of binary i e two scale media model problems moreover numerical experiments show that the new preconditioner has greatly improved performance over standard preconditioners even in the random coefficient case we show also how the analysis extends in a straightforward way to multiplicative versions of the schwarz method

two dimensional exponential fitting and applications to drift diffusion models a class of methods is introduced that generalizes the so called exponential fitting method to two dimensional problems and more specifically to equations of the type operatorname div underset raise0 3em hbox smash scriptscriptstyle nabla u u underset raise0 3em hbox smash scriptscriptstyle nabla psi f with psi given and underset raise0 3em hbox smash scriptscriptstyle nabla psi big in a part of the domain the basic idea is the following i write the equation as operatorname div e psi underset raise0 3em hbox smash scriptscriptstyle nabla rho f through the change of variables u rho e psi ii discretize the symmetric equation by means of a method that uses some kind of harmonic average for the coefficient e psi mixed hybrid etc iii write the discrete scheme in terms of the nodal values of u e psi rho this produces an m matrix

numerical algorithms based on galerkin methods for the modeling of reactive interfaces in photoelectrochemical pec solar cells abstract this work concerns the numerical solution of a coupled system of self consistent reaction drift diffusion poisson equations that describes the macroscopic dynamics of charge transport in photoelectrochemical pec solar cells with reactive semiconductor and electrolyte interfaces we present three numerical algorithms mainly based on a mixed finite element and a local discontinuous galerkin method for spatial discretization with carefully chosen numerical fluxes and implicit explicit time stepping techniques for solving the time dependent nonlinear systems of partial differential equations we perform computational simulations under various model parameters to demonstrate the performance of the proposed numerical algorithms as well as the impact of these parameters on the solution to the model

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100