discontinuous galerkin methods for friedrichs systems i general theory this paper presents a unified analysis of discontinuous galerkin methods to approximate friedrichs systems an abstract set of conditions is identified at the continuous level to guarantee existence and uniqueness of the solution in a subspace of the graph of the differential operator then a general discontinuous galerkin method that weakly enforces boundary conditions and mildly penalizes interface jumps is proposed all the design constraints of the method are fully stated and an abstract error analysis in the spirit of strang s second lemma is presented finally the method is formulated locally using element fluxes and links with other formulations are discussed details are given for three examples namely advection reaction equations advection diffusion reaction equations and the maxwell equations in the so called elliptic regime

an iterative scheme for a class of quasi variational inequalities an iterative scheme is given to obtain the approximate solution of a class of quasi variational inequalities it is shown that the approximate solution obtained by the iterative scheme converges strongly in the hilbert space to the exact solution as a special case we obtain the corresponding iterative scheme for variational inequalities

high order discrete helmholtz decompositions for the electric field integral equation we develop a differential form based formalism to address the problem of low frequency breakdown of the electric field integral equation efie note in this formalism we approximate the surface magnetic field not the surface current as is conventionally the case a discrete helmholtz decomposition is achieved for both triangular and quadrilateral curvilinear meshes based on a star cotree decomposition these decompositions are based upon the construction of a canonical basis which ab initio possess the required separation into irrotational and nonirrotational spaces this makes the process of construction clear and generally applicable the construction of appropriate bases is demonstrated for a range of interpolation orders the effects of these constructions is demonstrated on a simple flat pec plate problem

perturbed proximal point algorithms for generalized quasivariational inclusions abstract in this paper we study a class of generalized quasivariational inclusions by using the properties of the resolvent operator associated with a maximal monotone mapping in hilbert space we have established an existence theorem of solutions for generalized quasivariational inclusions suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized quasivariational inclusions as special cases some known results in this field are also discussed

algorithm for generalized multivalued variational inequalities in hilbert spaces abstract we introduce a new class of generalized multivalued variational inequality problems 1 1 which contain several existing variational inequality problems as special cases an iterative algorithm for finding approximate solutions of problem 1 1 is considered several convergence results for this algorithm are derived and in particular several existence results of problems 1 1 are obtained we also introduce a new class of generalized multivalued complementarity problems 5 1 which also contain several known complementarity problems as special cases it is shown that problem 5 1 is equivalent to problem 1 1 of special type from which several existence results of problem 5 1 are obtained

bayesian abstract fuzzy economies random quasi variational inequalities with random fuzzy mappings and random fixed point theorems abstract the main purpose of this paper is to introduce an abstract fuzzy economy model with differential asymmetric information and a measure space of agents it generalizes a previous model proposed by the author in 2009 the applications concern the fuzzy equilibrium existence and the existence of the solutions for two types of random quasi variational inequalities furthermore the paper presents random fixed point theorems with random fuzzy mappings extensions of the ones with random data our results either generalize or improve corresponding ones present in literature they consider the bochner integrability setting a measure space of indices and use random fuzzy mappings which describe the uncertainties

coincidence theorems and variational inequalities for fuzzy mappings abstract in this paper some coincidence theorems and existence theorems of solutions of variational inequalities for fuzzy mappings are established

explicit runge kutta schemes and finite elements with symmetric stabilization for first order linear pde systems we analyze explicit runge kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first order linear differential operator in space of friedrichs type for the time discretization we consider explicit second and third order runge kutta schemes we identify a general set of properties on the space stabilization encompassing continuous and discontinuous finite elements under which we prove stability estimates using energy arguments then we establish l 2 norm error estimates with quasi optimal convergence rates for smooth solutions in space and time these results hold under the usual cfl condition for third order runge kutta schemes and any polynomial degree in space and for second order runge kutta schemes and first order polynomials in space for second order runge kutta schemes and higher polynomial degrees in space a tightened 4 3 cfl condition is required numerical results are presented for smooth and rough solutions the case of finite volumes is briefly discussed

convergence and stability of perturbed three step iterative algorithm for completely generalized nonlinear quasivariational inequalities in this paper we introduce a new class of completely generalized nonlinear quasivariational inequalities and obtain its equivalence with a class of fixed point problems by using the resolvent operator technique using this equivalence we develop perturbed three step iterative algorithm for this class of completely generalized quasivariational inequalities we establish a few existence theorems of unique solution for the class of completely generalized quasivariational inequalities involving relaxed monotone generalized pseudocontractive and strongly monotone mappings and prove some convergence and stability results of iterative sequence generated by perturbed three step iterative algorithm

discontinuous galerkin methods for friedrichs systems part iii multifield theories with partial coercivity this paper is the third and last part of a work attempting to give a unified analysis of discontinuous galerkin methods the purpose of this paper is to extend the framework that has been developed in part ii for two field friedrichs systems associated with second order pdes we now consider two field friedrichs systems with partial l 2 coercivity and three field friedrichs systems with an even weaker l 2 coercivity hypothesis in particular this work generalizes the discontinuous galerkin methods of part ii to compressible and incompressible linear continuum mechanics we also show how the stabilizing parameters of the method must be set when the two field friedrichs system is composed of terms that may be of different magnitude thus accounting for instance for advection diffusion equations at high peclet numbers

friedrichs systems in a hilbert space framework solvability and multiplicity abstract the friedrichs 1958 theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics irrespective of their type this theory was recast in an abstract hilbert space setting by ern guermond and caplain 2007 and by antonic and burazin 2010 in this work we make a further step presenting a purely operator theoretic description of abstract friedrichs systems and proving that any pair of abstract friedrichs operators admits bijective extensions with a signed boundary map moreover we provide sufficient and necessary conditions for existence of infinitely many such pairs of spaces and by the universal operator extension theory grubb 1968 we get a complete identification of all such pairs which we illustrate on two concrete one dimensional examples

strongly nonlinear quasivariational inequalities abstract in this paper we develop the algorithms for finding the approximate solution of a strongly nonlinear quasivariational inequality and a strongly nonlinear quasicomplementarity problem and we include as special cases known results in this area we also observe that these two problems are equivalent if the convex set involved in the formulation of these problem is a convex cone

a perturbed algorithm for variational inclusions a perturbed method for solving a new class of variational inclusions is presented and a convergence result which includes as a special case some known results in this field is given

an iterative algorithm for variational inequalities abstract in this paper we introduce and study a new class of nonlinear variational inequalities this new class enables us to apply variational techniques to the solution of differential equations of both odd and even orders a projection method is used to suggest an iterative algorithm for finding the approximate solution of this class we also discuss the convergence criteria of the proposed iterative algorithm several special cases are discussed which can be obtained from the general result

a posteriori error analysis for numerical approximations of friedrichs systems the global error of numerical approximations for symmetric positive systems in the sense of friedrichs is decomposed into a locally created part and a propagating component residual based two sided local a posteriori error bounds are derived for the locally created part of the global error these suggest taking the l 2 norm as well as weaker dual norms of the computable residual as local error indicators the dual graph norm of the residual vec r h is further bounded from above and below in terms of the l 2 norm of h vec r h where h is the local mesh size the theoretical results are illustrated by a series of numerical experiments

boundary operator from matrix field formulation of boundary conditions for friedrichs systems abstract following the recent progress in understanding the abstract setting for friedrichs symmetric positive systems by ern guermond and caplain 2007 8 as well as antonic and burazin 2010 3 an attempt is made to relate these results to the classical friedrichs theory a comparison of two approaches via the trace operator and the boundary operator has been made favouring the latter finally a particular set of sufficient conditions for a boundary matrix field to define a boundary operator in that case is given and the applicability of this procedure in realistic situations is shown by examples

on random variational inclusions with random fuzzy mappings and random relaxed cocoercive mappings in this paper we introduce and study the random variational inclusions with random fuzzy and random relaxed cocoercive mappings we define an iterative algorithm for finding the approximate solutions of this class of variational inclusions and establish the convergence of iterative sequences generated by proposed algorithm our results improve and generalize many known corresponding results

a couple of nonlinear equations with fuzzy mappings in fuzzy normed spaces in this paper we introduce the concept of f contractor couple and study the existence of solutions for a couple of nonlinear equations with fuzzy mappings in fuzzy normed spaces we construct some new iterative algorithms for solving the couple of nonlinear equations with fuzzy mappings and prove the convergence of iterative sequences generated by the algorithms in fuzzy normed spaces as applications we obtain some new common fixed point theorems for fuzzy mappings in fuzzy normed spaces the results presented in this paper improve and generalize corresponding results of cho et al fuzzy sets and systems 110 2000 115 and fang and song fuzzy sets and systems 121 2001 267

generalized quasi variational like inclusions with nonconvex functionals in this paper two new concepts of h subdifferential and h proximal mappings of a proper functional is introduced in hilbert spaces the existence and lipschitz continuity of h proximal mapping of a proper functional are proved by applying these concepts we introduce and study a class of generalized quasi variational like inclusions with a nonconvex functional a new iterative algorithms for finding the approximate solutions and the convergence criteria of the iterative sequences generated by the algorithm are also given these algorithm and existence result generalize many known results in the literature

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

a unified discontinuous petrov galerkin method and its analysis for friedrichs systems we propose a unified discontinuous petrov galerkin dpg framework with optimal test functions for friedrichs like systems which embrace a large class of elliptic parabolic and hyperbolic partial differential equations pdes the well posedness i e existence uniqueness and stability of the dpg solution is established on a single abstract dpg formulation and two abstract dpg methods corresponding to two different but equivalent norms are devised we then apply the single dpg framework to several linear ized pdes including but not limited to scalar transport laplace diffusion convection diffusion convection diffusion reaction linear ized continuum mechanics e g linear ized elasticity a version of linearized navier stokes equations etc time domain acoustics and a version of the maxwell s equations the results show that we not only recover several existing dpg methods but also discover new dpg methods for both pdes currently considered in the dpg community and new ones as

high order calderon preconditioned time domain integral equation solvers two high order accurate calderon preconditioned time domain electric field integral equation tdefie solvers are presented in contrast to existing calderon preconditioned time domain solvers the proposed preconditioner allows for high order surface representations and current expansions by using a novel set of fully localized high order div and quasi curl conforming dqcc basis functions numerical results demonstrate that the linear systems of equations obtained using the proposed basis functions converge rapidly regardless of the mesh density and of the order of the current expansion

an approach for solving fuzzy implicit variational inequalities with linear membership functions in this paper we consider a class of fuzzy implicit variational inequalities with linear membership functions by using the tolerance approach we show that solving such problems can be reduced to a semi infinite programming problem a version of the method of centres with entropic regularization techniques only used a quasi newton line search using matlab software is required in our implementation we also give a numerical example to illustrate the validity of our approach

implicit runge kutta methods for the discretization of time domain integral equations implicit runge kutta based schemes are proposed for solving the time domain integral equations of electromagnetic theory the proposed technique maps a laplace domain equation to a domain equation using the butcher tableau of the implicit runge kutta scheme a discrete time domain system is recovered by computing the inverse transform numerically the resulting technique is capable of third or fifth order accuracy in time and is stable independent of time step numerical results illustrate the accuracy and stability of the technique

generalized nonlinear random a η accretive equations with random relaxed cocoercive mappings in banach spaces in this paper we introduce and study a new class of generalized nonlinear random a h accretive equations with random relaxed cocoercive mappings in banach spaces by using the chang s lemma and the resolvent mapping technique for a h accretive mappings due to lan et al h y lan y j cho r u verma nonlinear relaxed cocoercive variational inclusions involving a h accretive mappings in banach spaces comput math appl 51 2006 1529 1538 we also prove the existence theorems of the solution and convergence theorems of the generalized random iterative procedures with errors for these nonlinear random equations in q uniformly smooth banach spaces the results presented in this paper improve and generalize some known corresponding results in the literature

generalized complementarity problems for fuzzy mappings in this paper we introduce a new class of generalized complementarity problems for the fuzzy mappings and construct a new iterative algorithm we also discuss the existence of solutions for the generalized complementarity problems and the convergence of iterative sequence

heat equation as a friedrichs system abstract inspired by recent advances in the theory of friedrichs symmetric positive systems we apply newly developed results to the heat equation by showing how the intrinsic theory of ern guermond and caplain 2007 can be used in order to get a well posedness result for the dirichlet initial boundary value problem we also demonstrate the application of the two field theory with partial coercivity of ern and guermond 2008 originally developed for elliptic problems and also discuss different possibilities for the construction of the appropriate boundary operator

random generalized nonlinear variational inclusions for random fuzzy mappings in this paper we introduce and study a new class of random set valued nonlinear generalized variational inclusions with noncompact valued random fuzzy mappings and construct a new iterative algorithm we also discuss the existence of random solutions for this class of random fuzzy variational inclusions and the convergence of random iterative sequences generated by the algorithm

variational inequalities for fuzzy mappings ii in this paper we establish the equivalence between the variational inequalities for fuzzy mappings and the wiener hopf equations for fuzzy mappings we use this equivalence to suggest a number of new iterative algorithms for solving the variational inequalities for fuzzy mappings we also study the convergence criteria of these iterative algorithms the auxiliary principle technique is used to study the existence of a solution of the variational inequality for fuzzy mappings and to suggest a novel and innovative iterative algorithm for computing the approximate solution the results proved in this paper represent an improvement of previously known results

higher order frequency domain computational electromagnetics a review of the higher order computational electromagnetics cem for antenna wireless and microwave engineering applications is presented higher order cem techniques use current field basis functions of higher orders defined on large e g on the order of a wavelength in each dimension curvilinear geometrical elements which greatly reduces the number of unknowns for a given problem the paper reviews all major surface volume integral and differential equation electromagnetic formulations within a higher order computational framework focusing on frequency domain solutions with a systematic and unified review of generalized curved parametric quadrilateral triangular hexahedral and tetrahedral elements and various types of higher order hierarchical and interpolatory vector basis functions in both divergence and curl conforming arrangements a large number of actual higher order techniques representing various combinations of formulations elements bases and solution procedures are identified and discussed the examples demonstrate the accuracy efficiency and versatility of higher order techniques and their advantages over low order discretizations the most important one being a much faster higher order convergence of the solution it is demonstrated that both components of the higher order modeling namely higher order geometrical modeling and higher order current field modeling are essential for accurate and efficient cem analysis of general antenna scattering and microwave structures

generalized multivalued quasi variational inequalities in this paper we introduce and study a new class of variational inequalities known as generalized multivalued quasi variational inequalities this class is the most general and unifying one using essentially the projection technique we establish the equivalence between this class and the implicit wiener hopf equations this equivalence is used to suggest a number of iterative algorithms for solving the generalized multivalued quasi variational inequalities the convergence analysis of these algorithms is also studied the results proved in this paper represent a significant improvement and refinement of the previously known results in this area

an accurate broad band method of moments using higher order basis functions and tree loop decomposition a method is presented that combines the low frequency accuracy of tree loop decomposition with higher order basis functions to achieve an electric field integral equation based method of moments that is accurate over a wide band of frequencies in general the use of higher order divergence conforming bases introduces more solenoidal degrees of freedom than the standard rao wilton glisson zeroth order bases these solenoidal degrees of freedom are counted and identified and an efficient method to construct the required higher order loops is presented numerical results demonstrate that the technique allows the use of a single mesh to acquire scattering results over a wide band of frequencies using arbitrary order basis functions

on the nonlinear complementarity problem abstract we consider and study an algorithm for a new class of complementarity problems of finding u such that u 0 tu a u 0 u tu a u 0 where t is a continuous mapping and a is a diagonally nonlinear mapping from r n into itself furthermore it is proved that the approximate solution obtained by the iterative scheme converges to the exact solution special cases are also discussed

approximation solvability of nonlinear random a η resolvent operator equations with random relaxed cocoercive operators in this paper we introduce and study a new class of nonlinear random a h resolvent operator equations with random relaxed cocoercive operators in hilbert spaces by using chang s lemma theorem 3 1 of liu and li and the resolvent operator technique for a h monotone operators due to lan we also prove the existence theorems of the solutions and convergence theorems of the new generalized random iterative procedures with errors for this nonlinear random a h resolvent operator equations involving non monotone random set valued operators in hilbert spaces the results presented in this paper improve and generalize some known corresponding results in the literature

robust solution of time domain integral equations using loop tree decomposition and bandlimited extrapolation a stabilization method that eradicates low frequency instabilities in the solution of the time domain integral equations of electromagnetic scattering is presented the method uses the loop tree decomposition originally suggested for the solution of low frequency scattering problems via the method of moments specifically a temporally differentiated form of the pertinent integral equation is tested using tree basis functions and the undifferentiated form is tested using solenoidal basis functions the underlying solution method uses bandlimited interpolation functions blifs and the higher order divergence conforming vector bases of graglia et al for the temporal and spatial discretization of the integral equations respectively an extrapolation technique has been implemented to overcome the noncausality introduced into the system by the blifs numerical results will demonstrate the stability and accuracy of the proposed technique

generalized nonlinear mixed quasi variational inequalities abstract in this paper we introduce and study a new class of quasi variational inequalities which is called the generalized nonlinear set valued mixed quasi variational inequality using the resolvent operator technique for maximal monotone mapping we construct some new iterative algorithms for solving this class of generalized nonlinear set valued mixed quasi variational inequalities we prove the existence of solution for this kind of generalized nonlinear set valued mixed quasivariational inequalities without compactness and the convergence of iterative sequences generated by the algorithms we also discuss the convergence and stability of perturbed iterative algorithm for solving a class of generalized nonlinear mixed quasi variational inequalities

some aspects of variational inequalities abstract in this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence uniqueness computational properties various generalizations sensitivity analysis and their applications we also propose some open problems with sufficient information and references so that someone may attempt solution s in his her area of special interest we also include some new results which we have recently obtained

complex friedrichs systems and applications recently there has been a significant development of the abstract theory of friedrichs systems in hilbert spaces ern a guermond j l and caplain g commun partial differ equations 32 317 341 2007 and antonic n and burazin k commun partial differ equations 35 1690 1715 2010 and its applications to specific problems in mathematical physics however these applications were essentially restricted to real systems we check that the already developed theory of abstract friedrichs systems can be adjusted to the complex setting with some necessary modifications which allows for applications to partial differential equations with complex coefficients we also provide examples where the involved hilbert space is not the space of square integrable functions as it was the case in previous studies but rather its closed subspace or the space hs rd cr for real s this setting appears to be suitable for particular systems of partial differential equations such as the dirac system the di

discontinuous galerkin methods for friedrichs systems part ii second order elliptic pdes this paper is the second part of a work attempting to give a unified analysis of discontinuous galerkin methods the setting under scrutiny is that of friedrichs systems endowed with a particular 2 times 2 structure in which one unknown can be eliminated to yield a system of second order elliptic like pdes for the remaining unknown a general discontinuous galerkin method for approximating such systems is proposed and analyzed the key feature is that the unknown that can be eliminated at the continuous level can also be eliminated at the discrete level by solving local problems all the design constraints on the boundary operators that weakly enforce boundary conditions and on the interface operators that penalize interface jumps are fully stated examples are given for advection diffusion reaction linear continuum mechanics and a simplified version of the magnetohydrodynamics equations comparisons with well known discontinuous galerkin approximations for the poisson equation are presented

zeroth order complete discretizations of integral equation formulations involving conducting or dielectric objects at very low frequencies we present the self loop basis functions a divergence conforming set with zero charge density these basis functions allow a rearrangement of the linear linear basis functions set to overcome the low frequency breakdown we show the accuracy and stability at very low frequencies of the linear linear discretizations of i electric field integral equation for perfectly conducting objects and ii the poggio miller chang harrington wu tsai formulation for dielectric bodies

general strongly nonlinear variational inequalities abstract in this paper we develop iterative algorithms for finding approximate solutions for new classes of variational and quasivariational inequalities which include as a special case some known results in this field it is shown that the solutions of the iterative schemes converge to the exact solutions

electromagnetic scattering from dispersive dielectric scatterers using the finite difference delay modeling method a new method for solving the time domain integral equations of electromagnetic scattering from dispersive dielectric bodies is proposed the method is based on the finite difference delay modeling scheme which accomplishes the temporal discretization by substituting a z domain finite difference approximation for the continuous time laplace domain s parameter the inverse z transform used for computing the temporal kernels is calculated numerically using the trapezoidal rule and inverse discrete fourier transform arbitrary dispersive models can be simulated without extraneous convolution using the new technique the normal field equations are augmented to the standard tangential equations to solve the low frequency instability problem

existence results of a perturbed variational inequality with a fuzzy mapping abstract this paper is devoted to the existence of a perturbed variational inequality with a fuzzy mapping in a finite dimensional space we introduce two perturbation methods for a fuzzy mapping in one method it is perturbed by a nonlinear mapping and in the other it is perturbed by a vector under suitable coercivity conditions we show that the perturbed variational inequality with a fuzzy mapping has a solution when perturbed by both of the methods mentioned above the main results presented in this paper generalize and improve some known results

on the existence of branch points in the eigenvalues of the electric field integral equation operator in the complex frequency plane it is demonstrated by specific examples that eigenvalues of the electric field integral equation efie operator for finite extent objects in lossless media can have branch points in the complex frequency s plane specifically it is shown from the analysis of the spheroidal wave equation in the s plane that branch points are present in the eigenvalues of the integral operator associated with the scalar scattering problem from a perfectly conducting spheroid similarly it is concluded from the analysis of the mathieu s differential equation that branch points besides that associated with the infinite extent of the object exist in the s plane behavior of the eigenvalues of the efie operator for a perfectly conducting infinite elliptic cylinder a proof is given that branch point singularities cannot occur in the eigenvalues of efie operators for structures in which geometrical symmetry completely determines the eigenfunctions and hence they are frequency independent it is conjectured that branch points may always be present when sufficient object symmetry is lacking this conjecture is supported by the fact that branch points appear when a sphere is deformed into a spheroid or when a circular cylinder is deformed into an elliptic one an analogous phenomenon has been observed in circuit and transmission line problems for example it is shown that when the taper parameter of an exponential transmission line goes to zero the line becomes uniform and thus a symmetrical structure the branch points in the eigenvalues of the impedance matrix move away to infinity

on the generalized implicit quasivariational inequalities abstract in the present paper we introduce and study a new class of generalized nonlinear implicit quasivariational inequalities for set valued mappings and construct some new iterative algorithms we prove the existence of solutions for this class of generalized nonlinear implicit quasivariational inequalities for set valued mappings without compactness and the convergence of iterative sequences generated by the algorithms we also study a new perturbed iterative algorithm for solving a class of general strongly nonlinear quasivariational inequalities