an efficient monte carlo interior penalty discontinuous galerkin method for elastic wave scattering in random media abstract this paper develops and analyzes an efficient monte carlo interior penalty discontinuous galerkin mcip dg method for elastic wave scattering in random media the method is constructed based on a multi modes expansion of the solution of the governing random partial differential equations it is proved that the mode functions satisfy a three term recurrence system of partial differential equations pdes which are nearly deterministic in the sense that the randomness only appears in the right hand side source terms not in the coefficients of the pdes moreover the same differential operator applies to all mode functions a proven unconditionally stable and optimally convergent ip dg method is used to discretize the deterministic pde operator an efficient numerical algorithm is proposed based on combining the monte carlo method and the ip dg method with the l u direct linear solver it is shown that the algorithm converges optimally with respect to both the mesh size h and the sampling number m and practically its total computational complexity only amounts to solving a few deterministic elastic helmholtz equations using a gaussian elimination direct linear solver numerical experiments are also presented to demonstrate the performance and key features of the proposed mcip dg method

modeling uncertainty in steady state diffusion problems via generalized polynomial chaos abstract we present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs in particular we focus on the solution of the poisson equation with random diffusivity forcing and boundary conditions the stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the askey scheme as a generalization of the original polynomial chaos idea of wiener amer j math 60 1938 897 a galerkin projection in random space is applied to derive the equations in the weak form the resulting set of deterministic equations for each random mode is solved iteratively by a block gauss seidel iteration technique both discrete and continuous random distributions are considered and convergence is verified in model problems and against monte carlo simulations

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

on finding and verifying locally optimal solutions the problem of finding locally optimal solutions to combinatorial problems in the framework of pls as defined by d s johnson c h papadimitriou and m yannakakis j comput system sci 37 1988 pp 79 100 is considered a pls complete problem is exhibited such that the problem of verifying local optimality can be solved in logspace for all previously known pls complete problems verifying local optimality was p complete and it was conjectured in j comput system sci 37 1988 pp 79 100 that this was necessary

identification of time and space dependent relaxation kernels for materials with memory related to cylindrical domains ii abstract in this paper we prove the main abstract results stated in 2 taking advantage of such results we prove the existence and uniqueness of a time and space dependent relaxation kernel in an integrodifferential parabolic equation related to cylindrical domains of the form a b 2 2being a smooth enough domain inrn

discontinuous galerkin methods for the helmholtz equation with large wave number this paper develops and analyzes some interior penalty discontinuous galerkin ipdg methods using piecewise linear polynomials for the helmholtz equation with the first order absorbing boundary condition in two and three dimensions it is proved that the proposed discontinuous galerkin methods are stable hence well posed without any mesh constraint for each fixed wave number k optimal order with respect to h error estimate in the broken h 1 norm and suboptimal order estimate in the l 2 norm are derived without any mesh constraint the latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime i e k 2 h gtrsim 1 numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect with respect to k in the error bounds the novelties of the proposed ipdg methods include the following first the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives second the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters since the helmholtz problem is a non hermitian and indefinite linear problem as expected the crucial and the most difficult part of the whole analysis is to establish the stability estimates i e a priori estimates for the numerical solutions to this end the cruxes of our analysis are to establish and to make use of a local version of the rellich identity for the laplacian and to mimic the stability analysis for the pde solutions given in p cummings analysis of finite element based numerical methods for acoustic waves elastic waves and fluid solid interactions in the frequency domain ph d thesis the university of tennessee knoxville tn 2001 p cummings and x feng math models methods appl sci 16 2006 pp 139 160 u hetmaniuk commun math sci 5 2007 pp 665 678

a priori and posteriori error analysis for time dependent maxwell s equations abstract we consider time dependent maxwell s equations discretized by variable time steps in time domain and edge elements in spatial domain first the stability and optimal a priori error estimate are proved for both semi and fully discrete schemes then a posteriori error analysis is carried out for both schemes

simple local search problems that are hard to solve many algorithms for np hard optimization problems find solutions that are locally optimal in the sense that the solutions cannot be improved by a polynomially computable perturbation very little is known about the complexity of finding locally optimal solutions either by local search algorithms or using other indirect methods johnson papadimitriou and yannakakis j comput system sci 37 1988 pp 79 100 studied this question by defining a complexity class pls that captures local search problems it was proved that finding a partition of a graph that is locally optimal into equal parts with respect to the acclaimed kernighan lin algorithm is pls complete it is shown here that several natural simple local search problems are pls complete and thus just as hard two examples are finding a partition that cannot be improved by a single swap of two vertices and finding a stable configuration for an undirected connectionist network when edges or other objects are unweighted then a local optimum

a projection based error analysis of hdg methods we introduce a new technique for the error analysis of hybridizable discontinuous galerkin hdg methods the technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods this renders the analysis of the projections of the discretization errors simple and concise by showing that these projections of the errors are bounded in terms of the distance between the solution and its projection our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection we illustrate the technique on a specific hdg method applied to a model second order elliptic problem

a priori and computable a posteriori error estimates for an hdg method for the coercive maxwell equations abstract in this paper we present and analyze a hybridizable discontinuous galerkin hdg method for a mixed curl curl formulation of the steady state coercive maxwell equations with a discrete sobolev embedding type estimates for the discontinuous polynomials we provide a priori error estimates for the electric field and the lagrange multiplier in the energy norm with the smooth or minimal regularity assumption on the exact solution we have optimal convergence rate for the electric field and the lagrange multiplier in the energy norm the a priori error estimate for the electric field in the l 2 norm is also obtained by the duality argument and the approximation is also optimal for the electric field in the l 2 norm moreover by employing suitable helmholtz decompositions of the error together with the upper bound estimate for the lagrange multiplier we provide a computable residual based a posteriori error estimator which is derived based on the error measured in terms of a mesh dependent energy norm the efficiency of the a posteriori error estimator is also established three dimensional numerical results testing the performance of the a priori and a posteriori error estimates for the maxwell equations are given

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

mixed discontinuous galerkin approximation of the maxwell operator we introduce and analyze a discontinuous galerkin discretization of the maxwell operator in mixed form here all the unknowns of the underlying system of partial differential equations are approximated by discontinuous finite element spaces of the same order for piecewise constant coefficients the method is shown to be stable and optimally convergent with respect to the mesh size numerical experiments highlighting the performance of the proposed method for problems with both smooth and singular analytical solutions are presented

stabilized interior penalty methods for the time harmonic maxwell equations we propose stabilized interior penalty discontinuous galerkin methods for the indefinite time harmonic maxwell system the methods are based on a mixed formulation of the boundary value problem chosen to provide control on the divergence of the electric field we prove optimal error estimates for the methods in the special case of smooth coefficients and perfectly conducting boundary using a duality approach

on the complexity of local search in unconstrained quadratic binary optimization we consider the problem of finding a local minimum of a binary quadratic function and show by an elementary construction that every descending local search algorithm takes exponential time in the worst case

quasilinear systems with memory kernels we consider a quasilinear integrodifferential system in non normal form such a system is a generalization of a phase field model with memory and includes as a particular case the system describing the combustion of a material with memory in this paper we study both the direct and the inverse problems our fundamental tools are the theory of analytic semigroups optimal regularity results and fixed point arguments

an absolutely stable discontinuous galerkin method for the indefinite time harmonic maxwell equations with large wave number this paper develops and analyzes an interior penalty discontinuous galerkin ipdg method using piecewise linear polynomials for the indefinite time harmonic maxwell equations with the impedance boundary condition in the three dimensional space the main novelties of the proposed ipdg method include the following first the method penalizes not only the jumps of the tangential component of the electric field across the element faces but also the jumps of the tangential component of its vorticity field second the penalty parameters are taken as complex numbers of negative imaginary parts for the differential problem we prove that the sesquilinear form associated with the maxwell problem satisfies a generalized weak stability i e inf sup condition for star shaped domains such a generalized weak stability readily infers wave number explicit a priori estimates for the solution of the maxwell problem which plays an important role in the error analysis for the ipdg method for the proposed ipdg metho

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

hybridizable discontinuous galerkin methods for the time harmonic maxwell s equations we present two hybridizable discontinuous galerkin hdg methods for the numerical solution of the time harmonic maxwell s equations the first hdg method explicitly enforces the divergence free condition and thus necessitates the introduction of a lagrange multiplier it produces a linear system for the degrees of freedom of the approximate traces of both the tangential component of the vector field and the lagrange multiplier the second hdg method does not explicitly enforce the divergence free condition and thus results in a linear system for the degrees of freedom of the approximate trace of the tangential component of the vector field only for both hdg methods the approximate vector field converges with the optimal order of k 1 in the l 2 norm when polynomials of degree k are used to represent all the approximate variables we propose elementwise postprocessing to obtain a new h c u r l conforming approximate vector field which converges with order k 1 in the h c u r l norm we present extensive numerical examples to demonstrate and compare the performance of the hdg methods

direct and inverse problems for a phase field model with memory we consider a semilinear integrodifferential system in non normal form such a system is a generalization of the one that arises in the phase field theory with memory we prove an abstract existence and uniqueness theorem and a continuous dependence result for the direct problem reformulating the direct problem in a suitable way we prove that the identification problem admits a unique solution

an inverse problem for a parabolic integrodifferential model in the theory of combustion abstract in this paper we investigate an abstract inverse problem that can be applied to the evolution equation u t t x u t x 0 t k t s u s x d s u t t x d x f u t x u t x given suitable initial boundary conditions here f is a given function and in the case f u t x u t x e u t x the evolution equation has applications in the theory of combustion since we identify the convolution memory kernel k and the temperature u we associate an additional measurement on the temperature of type x u t x d x g t where and g are given functions the novelty with respect to the existing literature is the presence of the term u t t x d x in the evolution equation that is motivated by a model in the theory of combustion we prove a local in time existence theorem and a global in time uniqueness result

an a posteriori error estimate for the variable degree raviart thomas method we propose a new a posteriori error analysis of the variable degree hybridized version of the raviart thomas method for second order elliptic problems on conforming meshes made of simplexes we establish both the reliability and eciency of the estimator for the l2 norm of the error of the ux we also nd the explicit dependence of the estimator on the order of the local spaces k 0 the only constants that are not explicitly computed are those depending on the shape regularity of the simplexes in particular the constant of the local eciency in equality is proven to behave like k 2 3 2 however we present numer ical experiments suggesting that such constant is actually independent of k

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

convergence and quasi optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems recently an adaptive nonconforming finite element method anfem has been developed by carstensen and hoppe in numer math 103 251 266 2006 in this paper we extend the result to some nonsymmetric and indefinite problems the main tools in our analysis are a posteriori error estimators and a quasi orthogonality property in this case we need to overcome two main difficulties one stems from the nonconformity of the finite element space the other is how to handle the effect of a nonsymmetric and indefinite bilinear form an appropriate adaptive nonconforming finite element method featuring a marking strategy based on the comparison of the a posteriori error estimator and a volume term is proposed for the lowest order crouzeix raviart element it is shown that the anfem is a contraction for the sum of the energy error and a scaled volume term between two consecutive adaptive loops moreover quasi optimality in the sense of quasi optimal algorithmic complexity can be shown for the anfem the results of numerical experiments confirm the theoretical findings

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

robust a posteriori error estimation for the maxwell equations we consider a robust a posteriori error estimator of residual type for the maxwell equations in a bounded two and three dimensional domain the continuous problem is approximated using conforming approximated spaces the main goal is to express the lower and upper bounds with respect to appropriate norms for that purpose a new interpolant of clement nedelec type is introduced and some interpolation error estimates are proved numerical tests are presented which confirm our theoretical results

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

a stochastic collocation method for elliptic partial differential equations with random input data this work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms these input data are assumed to depend on a finite number of random variables the method consists of a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it treats easily a wide range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction of the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method finally we include a section with developments posterior to the original publication of this work there we review sparse grid stochastic collocation methods which are effective collocation strategies for problems that depend on a moderately large number of random variables

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

breaking spaces and forms for the dpg method and applications including maxwell equations discontinuous petrov galerkin dpg methods are made easily implementable using broken test spaces i e spaces of functions with no continuity constraints across mesh element interfaces broken spaces derivable from a standard exact sequence of first order unbroken sobolev spaces are of particular interest a characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces this technique is used to provide a complete error analysis of dpg methods for maxwell equations with perfect electric boundary conditions the technique also permits considerable simplifications of previous analyses of dpg methods for other equations reliability and efficiency estimates for an error indicator also follow finally the equivalence of stability for various formulations of the same maxwell problem is proved including the strong form the ultraweak form and various forms in between

convergence and optimality of adaptive edge finite element methods for time harmonic maxwell equations we consider a standard adaptive edge finite element method aefem based on arbitrary order nedelec edge elements for three dimensional indefinite time harmonic maxwell equations we prove that the aefem gives a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops provided the initial mesh is fine enough using the geometric decay we show that the aefem yields the best possible decay rate of the error plus oscillation in terms of the number of degrees of freedom the main technical contribution of the paper is in the establishment of a quasi orthogonality and a localized a posteriori error estimator

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

a convergent nonconforming adaptive finite element method with quasi optimal complexity in this paper we prove convergence and quasi optimal complexity of a simple adaptive nonconforming finite element method in each step of the algorithm the iterative solution of the discrete system is controlled by an adaptive stopping criterion and the local refinement is based on either a simple edge residual or a volume term depending on an adaptive marking strategy we prove that this marking strategy guarantees a strict reduction of the error augmented by the volume term and an additional oscillation term and quasi optimal complexity of the generated sequence of meshes

discrete functional analysis tools for discontinuous galerkin methods with application to the incompressible navier stokes equations two discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes i a discrete counterpart of the continuous sobolev embeddings in both hilbertian and non hilbertian settings ii a compactness result for bounded sequences in a suitable discontinuous galerkin norm together with a weak convergence property for some discrete gradients the proofs rely on techniques inspired by the finite volume literature which differ from those commonly used in finite element analysis the discrete functional analysis tools are used to prove the convergence of discontinuous galerkin approximations of the steady incompressible navier stokes equations two discrete convective trilinear forms are proposed a non conservative one relying on temam s device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure

an adaptive multilevel method for time harmonic maxwell equations with singularities we develop an adaptive edge finite element method based on reliable and efficient residual based a posteriori error estimates for low frequency time harmonic maxwell equations with singularities the resulting discrete problem is solved by the multigrid preconditioned minres minimum residual iteration algorithm we demonstrate the efficiency and robustness of the proposed method by extensive numerical experiments for cavity problems with singular solutions which include in particular scattering over screens

a posteriori error estimation and adaptive mesh refinement techniques abstract we analyse three different a posteriori error estimators for elliptic partial differential equations they are based on the evaluation of local residuals with respect to the strong form of the differential equation on the solution of local problems with neumann boundary conditions and on the solution of local problems with dirichlet boundary conditions we prove that all three are equivalent and yield global upper and local lower bounds for the true error thus adaptive mesh refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities some numerical examples prove the efficiency of the error estimators and the mesh refinement techniques