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

on the characterization of probabilities a generalization of bell s inequalities a partial solution to the problem of generalizing bell s inequalities to arbitrary numbers of physical properties is proposed it is first assumed that the considered sets of probabilities correspond to events which satisfy a postulate ensuring that they form an orthomodular partially ordered set admitting a full set of states in this framework a theorem generalizing bell s inequalities to an arbitrary finite number of events is proven an interpretation of these results in hilbert space is indicated conditions characterizing the classicality of so called correlation probabilities are then found and a method for verifying inequalities involving measurable correlations is discussed

spatially balanced sampling of natural resources the spatial distribution of a natural resource is an important consideration in designing an efficient survey or monitoring program for the resource generally sample sites that are spatially balanced that is more or less evenly dispersed over the extent of the resource are more efficient than simple random sampling we review a unified strategy for selecting spatially balanced probability samples of natural resources the technique is based on creating a function that maps two dimensional space into one dimensional space thereby defining an ordered spatial address we use a restricted randomization to randomly order the addresses so that systematic sampling along the randomly ordered linear structure results in a spatially well balanced random sample variable inclusion probability proportional to an arbitrary positive ancillary variable is easily accommodated the basic technique selects points in a two dimensional continuum but is also applicable to sampling finite populations or one dimension

on the implementation of mixed methods as nonconforming methods for second order elliptic problems in this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation lower order terms are treated so our results apply also to parabolic equations we define an approximation method by incorporating some projection operators within a standard galerkin method which we call a projection finite element method it is shown that for a given mixed method if the projection method s finite element space m h satisfies three conditions then the two approximation methods are equivalent these three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator we then construct appropriate nonconforming spaces m h for the known triangular and rectangular elements the lowest order raviart thomas mixed solution on rectangular finite elements in r 2 and r 3 on simplices or on prisms is then implemented as a nonconforming method modified in a simple and computationally trivial manner this new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution a rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal order multigrid methods for the solution of the linear system

design based maps for continuous spatial populations summarywe analyse the estimation of values of a survey variable throughout a continuum of points in a study area when a sample of points is selected by a probabilistic sampling scheme at each point the value is estimated using an inverse distance weighting interpolator and maps of the survey variable can be obtained we investigate the design based asymptotic properties of the interpolator when the study area remains fixed and the number of sampled points approaches infinity and we derive conditions ensuring design based asymptotic unbiasedness and consistency the conditions essentially require the existence of a pointwise or uniformly continuous function describing the behaviour of the survey variable and the use of spatially balanced designs to select points finally we propose a computationally simple mean squared error estimator

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

design based maps for finite populations of spatial units abstractthe estimation of the values of a survey variable in finite populations of spatial units is considered for making maps when samples of spatial units are selected by probabilistic sampling schemes the single values are estimated by means of an inverse distance weighting predictor the design based asymptotic properties of the resulting maps referred to as the design based maps are considered when the study area remains fixed and the sizes of the spatial units tend to zero conditions ensuring design based asymptotic unbiasedness and consistency are derived they essentially require the existence of a pointwise or uniformly continuous density function of the survey variable onto the study area some regularities in the size and shape of the units and the use of spatially balanced designs to select units the continuity assumption can be relaxed into a riemann integrability assumption when estimation is performed at a sufficiently small spatial grain and the estimates are subsequently aggregated

superconvergence and extrapolation for mixed finite element methods on rectangular domains asymptotic expansions for the rt raviart thomas mixed finite element approximation by the lowest order rectangular element associated with a second order elliptic equation on a rectangular domain are derived superconvergence for the vector field along the gauss lines is obtained as a result of the expansion a procedure of postprocessed extrapolation is presented for the scalar field as well as procedures of pure richardson extrapolation for both the vector and the scalar fields

superconvergence of both the crouzeix raviart and morley elements in this paper a new method is proposed to prove the superconvergence of both the crouzeix raviart and morley elements the main idea is to fully employ equivalences with the first order raviart thomas element and the first order hellan herrmann johnson element respectively in this way some special conformity of discrete stresses is explored and superconvergence of mixed elements can be used to analyze superconvergence of nonconforming elements finally the superconvergence of one and a half order by postprocessing is proved for both nonconforming elements

decoupling mixed finite elements on hierarchical triangular grids for parabolic problems in this paper we propose a numerical method for the solution of time dependent flow problems in mixed form such problems can be efficiently approximated on hierarchical grids obtained from an unstructured coarse triangulation by using a regular refinement process inside each of the initial coarse elements if these elements are considered as subdomains we can formulate a non overlapping domain decomposition method based on the lowest order raviart thomas elements properly enhanced with lagrange multipliers on the boundaries of each subdomain excluding the dirichlet edges a suitable choice of mixed finite element spaces and quadrature rules yields a cell centered scheme for the pressures with a local 10 point stencil the resulting system of differential algebraic equations is integrated in time by the crank nicolson method which is known to be a stiffly accurate scheme as a result we obtain independent subdomain linear systems that can be solved in parallel the behavior of the algorithm is illustrated on a variety of numerical experiments

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

a discontinuous petrov galerkin method with lagrangian multipliers for second order elliptic problems we present a discontinuous petrov galerkin dpg method for the finite element discretization of second order elliptic boundary value problems the novel approach emanates from a one element weak formulation of the differential problem this procedure which is typical of discontinuous galerkin dg methods is based on introducing variables defined in the interior and on the boundary of the element the interface variables are suitable lagrangian multipliers that enforce interelement continuity of the solution and of its normal derivative thus providing the proper connection between neighboring elements the internal variables can be eliminated in favor of the interface variables using static condensation to end up with a system of reduced size in the sole lagrangian multipliers a stability and convergence analysis of the novel formulation is carried out and its connection with mixed hybrid and dg methods is explored numerical tests on several benchmark problems are included to validate the convergence performance and the flux conservation properties of the dpg method

the patch test as a validation of a new finite element for the solution of convection diffusion equations abstract a first goal of this work is to show how the patch test a well known tool established in the field of continuum mechanics can be fruitfully employed to demonstrate the convergence and robustness of a numerical method for the solution of convection dominated flow problems a new finite element passing the patch test is then proposed for the discrestization of current continuity equations in semiconductors numerical results relative to the study of two dimensional devices are included to validate the effectiveness of the method

superconvergence for rectangular mixed finite elements in this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems for the rectangular finite elements of raviart and thomas 19 and for those of brezzi et al 4 we prove that the distance inl 2 between the approximate solution and a projection of the exact one is of higher order than the error itself this result is exploited to obtain superconvergence at gaussian points and to construct higher order approximations by a local postprocessing

error estimates for a finite element method for the drift diffusion semiconductor device equations in this paper optimal error estimates are obtained for a method for numerically solving the so called unipolar model a one dimensional simplified version of the drift diffusion semi conductor device equations the numerical method combines a mixed finite element method using a continuous piecewise linear approximation of the electric field with an explicit upwinding finite element method using a piecewise constant approximation of the electron concentration for initial and boundary data ensuring that the electron concentration is smooth the text l infty text l 1 error for the electron concentration and the text l infty text l infty error of the electric field are both proven to be of order delta x the error analysis is carried out first in the zero diffusion case in detail and then extended to the full unipolar model

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

discretization schemes on triangular grids abstract physical applications posed on irregular domains have caused difficulties in the use of many otherwise effective discretization schemes in two dimensional examples we consider discretization schemes on triangles to approximate irregular domains although finite element mixed finite element and finite volume schemes for triangles are well understood efficient solvers are essential for their effective use in this paper we consider certain cell centered finite difference schemes on triangles both for their theoretical properties on regular grids and as effective preconditioners for other discretization schemes for more general triangulations two cell centered finite difference schemes four point star stencils are derived from the simplification of the lowest order raviart thomas mixed finite element discretization on certain triangular grids optimal and superconvergence error estimates for both the solution and the gradient of the solution are established the accuracy of these schemes is also discussed in the case of discontinuous coefficients computational experiments verify the error estimates finally the use of these methods as preconditioners is also discussed

enhanced cell centered finite differences for elliptic equations on general geometry we present an expanded mixed finite element method for solving second order elliptic partial differential equations on geometrically general domains for the lowest order raviart thomas approximating spaces we use quadrature rules to reduce the method to cell centered finite differences possibly enhanced with some face centered pressures this substantially reduces the computational complexity of the problem to a symmetric positive definite system for essentially only as many unknowns as elements our new method handles general shape elements triangles quadrilaterals and hexahedra and full tensor coefficients while the standard mixed formulation reduces to finite differences only in special cases with rectangular elements as in other mixed methods we maintain the local approximation of the divergence i e local mass conservation in contrast galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation our method is shown to be as accurate as the standard mixed method for a large class of smooth meshes on nonsmooth meshes or with nonsmooth coefficients one can add lagrange multiplier pressure unknowns on certain element edges or faces this enhanced cell centered procedure recovers full accuracy with little additional cost if the coefficients or mesh geometry are piecewise smooth theoretical error estimates and numerical examples are given illustrating the accuracy and efficiency of the methods

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

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

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

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

a new mass lumping scheme for the mixed hybrid finite element method diffusion type partial differential equation is a common mathematical model in physics solved by mixed finite elements it leads to a system matrix which is not always an m matrix therefore the numerical solution may exhibit unphysical results due to oscillations the criterion necessary to obtain an m matrix is discussed in details for triangular rectangular and tetrahedral elements it is shown that the system matrix is never an m matrix for rectangular elements and can be an m matrix for triangular an tetrahedral elements if criteria on the element s shape and on the time step length are fulfilled a new mass lumping scheme is developed which leads to a less restrictive criterion the discretization must be weakly acute all angles less than 2 and there is no constraint on the time step length the lumped formulation of mixed hybrid finite element can be applied not only to triangular meshes but also to more general shape elements in two and three dimensions numerical experiments show that compared to the standard mixed hybrid formulation the lumping scheme avoids or strongly reduce oscillations and does not create additional numerical errors copyright 2006 john wiley sons ltd

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

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

superconvergence of mixed finite element approximations over quadrilaterals a superconvergence result is established in this article for approximate solutions of second order elliptic equations by mixed finite element methods over quadrilaterals the superconvergence indicates an accuracy of cal o h k 2 for the mixed finite element approximation if the raviart thomas or brezzi douglas fortin marini elements of order k are employed with optimal error estimate of cal o h k 1 numerical experiments are presented to illustrate the theoretical result

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

the bell phenomenon in classical frameworks the experimentally tested occurrence of probabilities that do not find a representation within the usual kolmogorov probability theory is mathematically formalized through the notion of the bell phenomenon for probability measures and for observables reference is made to a physically natural definition of observables that includes both the classical and the quantum usual versions it is shown that even a classical framework can host the bell phenomenon provided fuzzy observables are called into play

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

stability and error analysis of mixed finite volume methods for advection dominated problems abstract we consider a convection diffusion reaction problem and we analyze a stabilized mixed finite volume scheme introduced in 1 the scheme is presented in the format of discontinuous galerkin methods and error bounds are given proving o h convergence in the l 2 norm for the scalar variable which is approximated with piecewise constant elements

design based asymptotics for two phase sampling strategies in environmental surveys summary we analyse design based properties of two phase strategies for estimating totals and nonlinear functions of totals for environmental populations when the sampling schemes are uniquely determined by points placed in the study region in the first phase points are located using tessellation stratified sampling whereas in the second phase a finite population sampling scheme is adopted we give sufficient conditions on second phase designs that ensure consistency and we investigate the variance convergence rate for some familiar schemes

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

on a characterization of classical and nonclassical probabilities in this paper a comparison of various approaches to the problem of characterization of probabilities is presented the notion of multidimensional probability or s probability is introduced and it is shown that this notion can be profitably used for characterization of probability systems the main results are given in theorems 3 and 4 in theorem 3 the conditions are considered under which a system of s probabilities admits a model on a generalized event space while in theorem 4 the conditions under which the system admits a classical model are considered an interpretation for s probabilities is discussed and it is shown that theorems 3 and 4 allow one to derive the logical structure of events from the observed probabilities

a pde based partial discharge simulator abstract partial discharges are the main ageing and failure mechanism of solid insulating materials subjected to alternated current stresses this phenomenon from a simulation point of view has been almost always tackled using semi empirical schemes in this work a fully physically based model based on a set of conservation partial differential equations is introduced a numerical algorithm specifically designed to solve this particular problem is developed and its validation is discussed considering some experimental data acquired in a simple geometry containing an isolated void

a nonconforming finite element method for the biot s consolidation model in poroelasticity a stable finite element scheme that avoids pressure oscillations for a three field biot s model in poroelasticity is considered the involved variables are the displacements fluid flux darcy velocity and the pore pressure and they are discretized by using the lowest possible approximation order crouzeix raviart finite elements for the displacements lowest order raviart thomas nedelec elements for the darcy velocity and piecewise constant approximation for the pressure mass lumping technique is introduced for the raviart thomas nedelec elements in order to eliminate the darcy velocity and therefore reduce the computational cost we show convergence of the discrete scheme which is implicit in time and use these types of elements in space with and without mass lumping finally numerical experiments illustrate the convergence of the method and show its effectiveness to avoid spurious pressure oscillations when mass lumping for the raviart thomas nedelec elements is used

an a priori error estimate for a monotone mixed finite element discretization of a convection diffusion problem we present a local exponential fitting hybridized mixed finite element method for convection diffusion problem on a bounded domain with mixed dirichlet neuman boundary conditions with a new technique that interpretes the algebraic system after static condensation as a bilinear form acting on certain lifting operators we prove an a priori error estimate on the lagrange multipliers that requires minimal regularity while an extension of more classical arguments provide an estimate for the other solution components

analysis of a finite element method for the drift diffusion semiconductor device equations the multidimensional case an explicit finite element method for numerically solving the drift diffusion semiconductor device equations in two space dimensions is analyzed the method is based on the use of a mixed finite element method for the approximation of the electric field and a discontinuous upwinding finite element method for the approximation of the electron and hole concentrations the mixed method gives an approximate electric field in the precise form needed by the discontinuous method which is trivially conservative and fully parallelizable it is proven that the method produces uniformly bounded concentrations and electric fields and that it converges to the exact solution provided there is a convergent subsequence of the electron concentrations numerical simulations are presented that display the performance of the method and indicate the behavior of the solution

superconvergence of the velocity along the gauss lines in mixed finite element methods superconvergence for the velocity along the gauss lines is derived for the mixed finite element method for the second order elliptic equation in rectangular domains these estimates are based on the rectangular finite elements of raviart thomas type

newton method for reactive solute transport with equilibrium sorption in porous media we present a mass conservative numerical scheme for reactive solute transport in porous media the transport is modeled by a convection diffusion reaction equation including equilibrium sorption the scheme is based on the mixed finite element method mfem more precisely the lowest order raviart thomas elements and one step euler implicit the underlying fluid flow is described by the richards equation a possibly degenerate parabolic equation which is also discretized by mfem this work is a continuation of radu et al 2008 and radu et al 2009 1 2 where the algorithmic aspects of the scheme and the analysis of the discretization method are presented respectively here we consider the newton method for solving the fully discrete nonlinear systems arising on each time step after discretization the convergence of the scheme is analyzed in the case when the solute undergoes equilibrium sorption of freundlich type the problem becomes degenerate and a regularization step is necessary we derive sufficient conditions for the quadratic convergence of the newton scheme