some second derivative free variants of chebyshev halley methods in this paper we present some new variants of the chebyshev halley methods free from second derivative the new methods require two function and one first derivative evaluations per iteration and are shown to be third order convergent several numerical examples are given to illustrate the performance of the presented methods

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

a paradox in the approximation of dirichlet control problems in curved domains in this paper we study the approximation of a dirichlet control problem governed by an elliptic equation defined on a curved domain omega to solve this problem numerically it is usually necessary to approximate omega by a typically polygonal new domain omega h the difference between the solutions of both infinite dimensional control problems one formulated in omega and the second in omega h was studied in e casas and j sokolowski siam j control optim 48 2010 pp 3746 3780 where an error of order o h was proved in k deckelnick a gunther and m hinze siam j control optim 48 2009 pp 2798 2819 the numerical approximation of the problem defined in omega was considered the authors used a finite element method such that omega h was the polygon formed by the union of all triangles of the mesh of parameter h they proved an error of order o h 3 2 for the difference between continuous and discrete optimal controls here we show that the estimate obtained in e casas and j sokolowski siam j control optim 48 2010 pp 3746 3780 cannot be improved which leads to the paradox that the numerical solution is a better approximation of the optimal control than the exact one obtained just by changing the domain from omega to omega h

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

the steklov eigenvalue problem in a cuspidal domain in this paper we analyze the approximation by piecewise linear finite elements of a steklov eigenvalue problem in a plane domain with an external cusp this problem is not covered by the literature and its analysis requires a special treatment indeed we develop new trace theorems and we also obtain regularity results for the source counterpart moreover under appropriate assumptions on the meshes we present interpolation error estimates for functions in fractional sobolev spaces these estimates allow us to obtain appropriate convergence results of the source counterpart which in the context of the theory of compact operator are a fundamental tool in order to prove the convergence of the eigenpairs at the end we prove the convergence of the eigenpairs by using graded meshes and present some numerical tests

classroom note geometry and convergence of euler s and halley s methods we investigate the global convergence of euler s and halley s methods by using their geometric interpretation a generalization of these methods is also briefly discussed

a priori error estimates of the dtn fem for the transmission problem in acoustics abstract this paper is concerned with a variational approach solving the two dimensional acoustic transmission problems the original problem is reduced to an equivalent nonlocal boundary value problem by introducing an exact dirichlet to neumann dtn mapping in terms of fourier expansion series on an artificial boundary uniqueness and existence of solutions in appropriate sobolev spaces are established for the corresponding variational problem and its modification due to the truncation of dtn mapping a priori error estimates containing the effects of both element meshsize and truncation order of series for the finite element approximation are derived numerical experiments are also presented to illustrate the efficiency and accuracy of the numerical scheme

tensor nuclear norm lpv subspace identification linear parameter varying lpv subspace identification methods suffer from an exponential growth in number of parameters to estimate this results in problems with ill conditioning in literature attempts have been made to address the ill conditioning by using regularization its effectiveness hinges on suitable a priori knowledge in this paper we propose using a novel alternative regularization that is we first show that the lpv sub markov parameters can be organized into several tensors that are multilinear low rank by construction namely their matricization along any mode is a low rank matrix then we propose a novel convex method with tensor nuclear norm regularization which exploits this low rank property simulation results show that the novel method can have higher performance than the regularized lpv pbsid text opt technique in terms of variance accounted for

an adaptive strategy for elliptic problems including a posteriori controlled boundary approximation we derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary the numerical solution is assumed to be defined on a finite element mesh whose boundary vertices are located on the boundary of the continuous problem no assumption is made on a geometrically fitting shape a posteriori error estimates are given in the energy norm and the l 2 norm and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation furthermore a strategy is presented to compute the effect of the non discretized part of the domain on the error starting from a coarse mesh this especially implies that parts of the domain where the measured error is small stay non discretized the presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary the reliable computation of the a posteriori estimates and a mesh adaptation based on delaunay techniques numerical examples illustrate that errors outside the initial discretization will be detected

criteria for the approximation property for multigrid methods in nonnested spaces we extend the abstract frameworks for the multigrid analysis for nonconforming finite elements to the case where the assumptions of the second strang lemma are violated the consistency error is studied in detail for finite element discretizations on domains with curved boundaries this is applied to prove the approximation property for conforming elements stabilized q1 p0elements and nonconforming elements for linear elasticity on nonpolygonal domains proving the approximation property for the multigrid analysis for nonconforming finite element discretizations is formalized in 7 4 15 for many cases it suffices to verify criteria on the approximation quality and the consistency error in these papers it is required that a continuous bilinear form can be extended to a nonconforming finite element space which is not valid for many interesting applications the purpose of this paper is to establish a full set of criteria which guarantees the approximation property for a wide range of nonnested discretizations where we do not assume that the discrete bilinear form coincides with the continuous bilinear form for all conforming functions in the notation we follow bramble 5 chap 4 and our results can be applied directly to the multigrid theory described there the results extend known results by brenner 7 and stevenson 15 and they provide a systematic and constructive way of studying nonnested multigrid algorithms for more general nonnested spaces and varying forms the paper is organized as follows first we introduce an abstract setting describing a multigrid hierarchy for nonconforming discretizations of an elliptic partial differential equation without full regularity as usual the multigrid approximation property is derived by comparison with the finite element approximation property which we formulate using an interpolation operator and its adjoint with respect to the energy scalar product in a second step section 1 9 we derive the approximation property from consistency assumptions on a conforming comparison space similar to the approach in 7 in section 2 we consider the case of conforming finite elements on a polygonal approximation of the computational domain here we choose a comparison space consisting of curved finite elements after introducing a suitable interpolation operator we use the equivalence of the operator norm scale used throughout section 1 received by the editor january 23 2001 and in revised form march 21 2003 2000 mathematics subject classification primary 65n55 65f10

on chebyshev halley methods with sixth order convergence for solving non linear equations abstract in this paper we present a family of new variants of chebyshev halley methods the new methods have sixth order convergence although they only add one evaluation of the function at the point iterated by chebyshev halley methods the numerical results presented show that the new methods work better not only in order but in efficiency

some variants of chebyshev halley methods free from second derivative abstract in this paper we present some new modifications of chebyshev halley methods free from second derivative the convergence analysis shows that the new methods are third order convergent and require one function and two first derivative evaluations per iteration several numerical examples are given to illustrate the performance of the presented methods

subspace identification of bilinear and lpv systems for open and closed loop data in this paper we present a novel algorithm to identify lpv systems with affine parameter dependence operating under open and closed loop conditions a factorization is introduced which makes it possible to form a predictor that predicts the output which is based on past inputs outputs and scheduling data the predictor contains the lpv equivalent of the markov parameters using this predictor ideas from closed loop lti identification are developed to estimate the state sequence from which the lpv system matrices can be constructed a numerically efficient implementation is presented using the kernel method it turns out that if structure is present in the scheduling sequence the computational complexity reduces even more

numerical analysis of quasilinear parabolic equations under low regularity assumptions in this paper we carry out the numerical analysis of a class of quasilinear parabolic equations where the diffusion coefficient depends on the solution of the partial differential equation the goal is to prove error estimates for the fully discrete equation using discontinuous galerkin discretization in time dg 0 combined with piecewise linear finite elements in space this analysis is performed under minimal regularity assumptions on the data in particular we omit any assumption regarding existence of a second derivative in time of the solution

the second generation shifted boundary method and its numerical analysis abstract recently the shifted boundary method sbm was proposed within the class of unfitted or immersed or embedded finite element methods by reformulating the original boundary value problem over a surrogate approximate computational domain the sbm avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning accuracy is maintained by modifying the original boundary conditions using taylor expansions hence the name of the method that shifts the location and values of the boundary conditions in this article we present enhanced variational sbm formulations for the poisson and stokes problems with improved flexibility and robustness these simplified variational forms allow to relax some of the assumptions required by the mathematical proofs of stability and convergence of earlier implementations first we show that these new sbm implementations can be proved asymptotically stable and convergent even without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive second we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at dirichlet boundaries therefore avoiding the calibration of an additional stabilization parameter finally we prove enhanced l 2 estimates without the cumbersome assumption of earlier proofs that the surrogate domain is convex instead we rely on a conventional assumption that the boundary of the true domain is smooth which can also be replaced by requiring convexity of the true domain the aforementioned improvements open the way to a more general and efficient implementation of the shifted boundary method particularly in complex three dimensional geometries we complement these theoretical developments with numerical experiments in two and three dimensions

approximations of very weak solutions to boundary value problems standard weak solutions to the poisson problem on a bounded domain have square integrable derivatives which limits the admissible regularity of inhomogeneous data the concept of solution may be further weakened in order to define solutions when data is rough such as for inhomogeneous dirichlet data that is only square integrable over the boundary such very weak solutions satisfy a nonstandard variational form u v g v a galerkin approximation combined with an approximation of the right hand side g defines a finite element approximation of the very weak solution applying conforming linear elements leads to a discrete solution equivalent to the text book finite element solutionto the poisson problem in which the boundary data is approximated by l2 projections the l2 convergence rate of the discrete solution is o hs for some s in 0 1 2 that depends on the shape of the domain assuming a polygonal two dimensional or polyhedral three dimensional domain without slits and only square integr

modified chebyshev halley methods with sixth order convergence abstract in this paper we present a family of modified chebyshev halley methods the new methods add one evaluation of the function at the point iterated by chebyshev halley methods analysis of convergence shows that these methods have sixth order convergence numerical results show that the new methods can be of practical interest

letter to the editor on the global convergence of chebyshev s iterative method in a melman geometry and convergence of euler s and halley s methods siam rev 39 4 1997 728 735 the geometry and global convergence of euler s and halley s methods was studied now we complete melman s paper by considering other classical third order method chebyshev s method by using the geometric interpretation of this method a global convergence theorem is performed a comparison of the different hypothesis of convergence is also presented

subspace identification for non linear systems with measured input non linearities this paper uses subspace methods to identify a class of multi input multi output discrete time non linear time varying systems specifically we identify systems that are non linear in measured data and linear in unmeasured states numerical examples are presented to demonstrate the efficacy of the method

predictor based tensor regression pbtr for lpv subspace identification the major bottleneck in state of the art linear parameter varying lpv subspace methods is the curse of dimensionality during the first regression step in this paper the origin of the curse of dimensionality is pinpointed and subsequently a novel method is proposed which does not suffer from this bottleneck the problem is related to the lpv sub markov parameters these have inherent structure and are dependent on each other but state of the art lpv subspace methods parametrize the lpv sub markov parameters independently this means the inherent structure is not preserved in the parametrization in turn this leads to a superfluous parametrization with the curse of dimensionality the solution lies in using parametrizations which preserve the inherent structure sufficiently to avoid the curse of dimensionality in this paper a novel method based on tensor regression is proposed this novel method is named the predictor based tensor regression method pbtr this method preserves the inherent structure sufficiently to avoid the curse of dimensionality simulation results show that pbtr has superior performance with respect to both state of the art lpv subspace techniques and also non convex techniques

a discontinuous galerkin based immersed boundary method with non homogeneous boundary conditions and its application to elasticity we propose a discontinuous galerkin based immersed boundary method for elasticity problems the resulting numerical scheme does not require boundary fitting meshes and avoids boundary locking by switching the elements intersected by the boundary to a discontinuous galerkin approximation special emphasis is placed on the construction of a method that retains an optimal convergence rate in the presence of non homogeneous essential and natural boundary conditions the role of each one of the approximations introduced is illustrated by analyzing an analog problem in one spatial dimension finally extensive two and three dimensional numerical experiments on linear and nonlinear elasticity problems verify that the proposed method leads to optimal convergence rates under combinations of essential and natural boundary conditions

an improvement of the euler chebyshev iterative method abstract we present a new method for the computation of the solutions of nonlinear equations when it is necessary to use high precision we improve the euler chebyshev iterative method which is a generalization of an improvement of newton s method a symbolic computation allows us to find the best coefficients respect to the local order of convergence the adaptation of the strategy presented here gives an additional iteration function with an additional evaluation of the function it provides a lower cost if we use adaptive multi precision arithmetics the numerical results computed using this system with a floating point representing a maximum of 210 decimal digits support this theory

accelerated iterative methods for finding solutions of a system of nonlinear equations abstract in this paper we present a technique to construct iterative methods to approximate the zeros of a nonlinear equation f x 0 where f is a function of several variables this technique is based on the approximation of the inverse function of f and on the use of a fixed point iteration depending on the number of steps considered in the fixed point iteration or in other words the number of evaluations of the function f we obtain some variants of classical iterative processes to solve nonlinear equations these variants improve the order of convergence of classical methods finally we show some numerical examples where we use adaptive multi precision arithmetic in the computation that show a smaller cost

a family of multi point iterative methods for solving systems of nonlinear equations we extend to n dimensional case a known multi point family of iterative methods for solving nonlinear equations this family includes as particular cases some well known and also some new methods the main advantage of these methods is they have order three or four and they do not require the evaluation of any second or higher order frechet derivatives a local convergence analysis and numerical examples are provided

zero finder methods derived from obreshkov s techniques in this paper two families of zero finding iterative methods for solving nonlinear equations f x 0 are presented the key idea to derive them is to solve an initial value problem applying obreshkov like techniques more explicitly obreshkov s methods have been used to numerically solve an initial value problem that involves the inverse of the function f that defines the equation carrying out this procedure several methods with different orders of local convergence have been obtained an analysis of the efficiency of these methods is given finally we introduce the concept of extrapolated computational order of convergence with the aim of numerically test the given methods a procedure for the implementation of an iterative method with an adaptive multi precision arithmetic is also presented

finite element approximations in a non lipschitz domain part ii fil acosta gabriel universidad de buenos aires facultad de ciencias exactas y naturales departamento de matematica argentina

finite element approximation of spectral problems with neumann boundary conditions on curved domains this paper deals with the finite element approximation of the spectral problem for the laplace equation with neumann boundary conditions on a curved nonconvex domain convergence and optimal order error estimates are proved for standard piecewise linear continuous elements on a discrete polygonal domain h in tile framework of the abstract spectral approximation theory

on consistency of subspace methods for system identification subspace methods for identification of linear time invariant dynamical systems typically consist of two main steps first a so called subspace estimate is constructed this first step usually consists of estimating the range space of the extended observability matrix secondly an estimate of system parametersis obtained based on the subspace estimate in this paper the consistency of a large class of methods for estimating the extended observability matrix is analyzed persistence of excitation conditions on the input signal are given which guarantee consistent estimates for systems with only measurement noise for systems with process noise it is shown that a persistence of excitation condition on the input is not sufficient more precisely an example for which the subspace methods fail to give a consistent estimate of the transfer function is given this failure occurs even if the input is persistently exciting of any order it is also shown that this problem can be eliminated if stronger conditions on the input signal are imposed

adams like techniques for zero finder methods two families of zero finding iterative methods for nonlinear equations are presented we derive them solving an initial value problem using adams like multistep techniques namely adams methods have been used to solve the problem that consists in a differential equation in what appears the inverse function of the one which zero will be computed and the condition given by the value attained by it at the initial approximation performing this procedure several methods of different local orders of convergence have been obtained

isogeometric analysis on v reps first results abstract inspired by the introduction of volumetric modeling via volumetric representations v reps by massarwi and elber in 2016 in this paper we present a novel approach for the construction of isogeometric numerical methods for elliptic pdes on trimmed geometries seen as a special class of more general v reps we develop tools for approximation and local re parameterization of trimmed elements for three dimensional problems and we provide a theoretical framework that fully justifies our algorithmic choices we validate our approach both on two and three dimensional problems for diffusion and linear elasticity

sharp estimates for finite element approximations to elliptic problems with neumann boundary data of low regularity consider a homogeneous parabolic problem on a smooth bounded domain in n but with initial data and neumann boundary data of low regularity sharp interior maximum norm error estimates are given for a semidiscrete c0 finite element approximation to this problem these estimates are obtained by first establishing a new localized l estimate for semidiscrete finite element approximations on interior subdomains numerical examples illustrate the findings

identification of the deterministic part of mimo state space models given in innovations form from input output data the problem of linear multivariable state space model identification from input output data can under the presence of process and measurement noise be solved in a non iterative way when incorporating instrumental variables constructed from both input and output sequences in the recently developed class of multivariable output error state space model class of subspace model identification schemes abstract in this paper we describe two algorithms to identify a linear time invariant finite dimensional state space model from input output data the system to be identified is assumed to be excited by a measurable input and an unknown process tloise and the measurements are disturbed by unknown measurement noise both noise sequences are discrete zero mean white noise the first algorithm gives consistent estimates only for the case where the input also is zero mean white noise while the same result is obtained with the second algorithm without this constraint for the special case where the input signal is discrete zero mean white noise it is explicitly shown that this second algorithm is a special case of the recently developed multivariable output error state space moesp class of algorithms based on instrumental variables the usefulness of the presented schemes is highlighted in a realistic simulation study

modified families of newton halley and chebyshev methods this paper presents new families of newton type iterative methods newton halley and chebyshev methods for finding simple zero of univariate non linear equation permitting f x 0 in the vicinity of the root newton type iterative methods have well known geometric interpretation and admit their geometric derivation from a parabola these algorithms are comparable to the well known powerful classical methods of newton halley and chebyshev respectively and these can be seen as special cases of these families the efficiency of the presented methods is demonstrated by numerical examples

on finite element approximations of problems having inhomogeneous essential boundary conditions abstract the analysis and implementation of finite element methods for problems with inhomogeneous essential boundary conditions are considered the results are given for linear second order elliptic partial differential equations and for the nonlinear stationary navier stokes equations for certain easily implemented boundary treatments optimal error estimates and numerical examples are provided for problems posed on polyhedral domains

full linear multistep methods as root finders root finders based on full linear multistep methods lmms use previous function values derivatives and root estimates to iteratively find a root of a nonlinear function as ode solvers full lmms are typically not zero stable however used as root finders the interpolation points are convergent so that such stability issues are circumvented a general analysis is provided based on inverse polynomial interpolation which is used to prove a fundamental barrier on the convergence rate of any lmm based method we show using numerical examples that full lmm based methods perform excellently finally we also provide a robust implementation based on brents method that is guaranteed to converge

certain improvements of chebyshev halley methods with accelerated fourth order convergence abstract in this paper we present a new one parameter family of methods which improves the classical chebyshev halley methods with cubic convergence the convergence analysis shows that each method of the family is quartically convergent and per iteration it requires one evaluation of the given function and two of its first derivative the well known jarratt fourth order method is shown to be part of the family several numerical examples are given to illustrate the efficiency and performance of the presented methods

finite element approximations in a nonlipschitz domain in this paper we analyze the approximation by standard piecewise linear finite elements of a nonhomogeneous neumann problem in a cuspidal domain since the domain is not lipschitz many of the results on sobolev spaces which are fundamental in the usual error analysis do not apply therefore we need to work with weighted sobolev spaces and to develop some new theorems on traces and extensions we show that in the domain considered here suboptimal order can be obtained with quasi uniform meshes even when the exact solution is in h 2 and we prove that the optimal order with respect to the number of nodes can be recovered by using appropriate graded meshes

a priori error analysis for hdg methods using extensions from subdomains to achieve boundary conformity we present the first a priori error analysis of a technique that allows us to numerically solve steady state diffusion problems defined on curved domains by using finite element methods defined in polyhedral subdomains dh for a wide variety of hybridizable discontinuous galerkin and mixed methods we prove that the order of convergence in the l2 norm of the approximate flux and scalar unknowns is optimal as long as the distance between the boundary of the original domain and that of the computational domain h is of order h we also prove that the l 2 norm of a projection of the error of the scalar variable superconverges with a full additional order when the distance between and h is of order h 5 4 but with only half an additional order when such a distance is of order h finally we present numerical experiments confirming the theoretical results and showing that even when the distance between and h is of order h the above mentioned projection of the error of the scalar variable can still superconverge with a full additional order

galerkin approximations of abstract parabolic boundary value problems with rough boundary data l p theory etude des approximations de galerkin d un probleme aux valeurs limites parabolique abstrait avec des donnees aux limites grossieres determination des vitesses de convergence optimales dans les normes lp

consistency and relative efficiency of subspace methods we give a consistency proof for two subspace methods we then show the asymptotic equivalence of a special subspace method and the initial estimate proposed by hannan and rissanen finally a simulation study comparing two subspace methods and the maximum likelihood method is performed

a robust finite element method for nonhomogeneous dirichlet problems in domains with curved boundaries in this paper the authors consider a simple finite element method on an approximately polygonal domain using linear elements the dirichlet data are transferred in a natural way and the resulting linear system can be solved using multigrid techniques their analysis takes into account the change in domain and data transfer and optimal error estimates are obtained that are robust in the regularity of the boundary data provided they are at least square integrable it is proved that the natural extension of this finite element approximation to the original domain is optimal order accurate

asymptotic properties of subspace estimators since the proposal of subspace methods in the 1980s and early 1990s substantial efforts have been made in the analysis of the statistical properties of the algorithms this paper surveys the literature on the asymptotic properties of particular subspace methods used for linear dynamic time invariant discrete time systems the goals of this paper are threefold first this survey tries to present the most relevant results on the asymptotic properties of estimators obtained using subspace methods secondly the main methods and tools that have been used in the derivation of these results are presented to make the literature more accessible thirdly main unsolved questions and rewarding research topics are identified some of which can be attacked using the toolbox discussed in the paper

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

a hummel seebeck family of iterative methods with improved convergence and efficiency several improvements to a family of iterative methods deduced from a hummel seebeck theorem are presented a symbolic computation that allows to find the best coefficients respect to the local order of convergence is also given the theoretical and computational order of convergence for all methods is increased furthermore the efficiency of these methods applied to the functions tested is improved adapting the strategy presented here a new iteration function with a new evaluation of the function is obtained and using adaptive multi precision arithmetic a smaller cost is got the numerical results computed carrying out this procedure with a floating point system representing 1000 decimal digits support this theory

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

some new sixth order methods for solving non linear equations abstract in this paper we present some new sixth order methods based on the chebyshev halley methods per iteration the new methods require an additional evaluation of the function compared with chebyshev halley methods analysis of efficiency shows that the new methods can compete with newton s method and the classical third order methods as we show in numerical results

the optimal convergence rate of a c1 finite element method for non smooth domains we establish optimal up to arbitrary e 0 convergence rates for a finite element formulation of a model second order elliptic boundary value problem in a weightedh 2 sobolev space with 5th degree argyris elements this formulation arises while generalizing to the case of non smooth domains an unconditionally stable scheme developed by liu et al j g liu j liu r l pego stability and convergence of efficient navier stokes solvers via a commutator estimate comm pure appl math 60 2007 pp 1443 1487 for the navier stokes equations we prove the optimality for both quasiuniform and graded mesh refinements and provide numerical results that agree with our theoretical predictions

powell sabin splines with boundary conditions for polygonal and non polygonal domains powell sabin splines are piecewise quadratic polynomials with a global c 1 continuity defined on conforming triangulations imposing boundary conditions on such a spline leads to a set of constraints on the spline coefficients first we discuss boundary conditions defined on a polygonal domain before we treat boundary conditions on a general curved domain boundary we consider dirichlet and neumann conditions and we show that a particular choice of the ps triangles at the boundary can greatly simplify the corresponding constraints finally we consider an application where the techniques developed in this paper are used the numerical solution of a partial differential equation by the galerkin and collocation method

immersed finite element method for eigenvalue problem abstract we consider the approximation of elliptic eigenvalue problem with an interface the main aim of this paper is to prove the stability and convergence of an immersed finite element method ifem for eigenvalues using crouzeix raviart p 1 nonconforming approximation we show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem we analyze the ifem for elliptic eigenvalue problems with an interface and derive the optimal convergence of eigenvalues numerical experiments demonstrate our theoretical results

a locally modified parametric finite element method for interface problems we present a modified finite element method that is able to approximate interface problems with high accuracy we consider interface problems where the solution is continuous its derivatives however may be discontinuous across interface curves within the domain the proposed discretization is based on a local modification of the finite element basis functions using a fixed quadrilateral mesh instead of moving mesh nodes we resolve the interface locally by an adapted parametric approach all modifications are applied locally and in an implicit fashion the scheme is easy to implement and is well suited for time dependent moving interface problems we show optimal order of convergence for elliptic problems and further we give a bound on the condition number of the system matrix both estimates do not depend on the interface location relative to the mesh

laplace eigenvalues on regular polygons a series in 1 n abstract for regular polygons p n inscribed in a circle the eigenvalues of the laplacian converge as n to the known eigenvalues on a circle we compute the leading terms of n in a series in powers of 1 n by applying the calculus of moving surfaces to a piecewise smooth evolution from the circle to the polygon the o 1 n 2 term comes from hadamard s formula and reflects the change in area this term disappears if we transcribe the polygon scaling it to have the same area as the circle

new estimates for multilevel algorithms including the v cycle the purpose of this paper is to provide new estimates for certain multilevel algorithms in particular we are concerned with the simple additive multilevel algorithm discussed recently together with j xu and the standard v cycle algorithm with one smoothing step per grid we shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels even on nonconvex domains which do not provide full elliptic regularity for example the theory applies to the standard multigrid vcycle on the l shaped domain or a domain with a crack and yields a uniform convergence rate we also prove uniform convergence rates for the multigrid v cycle for problems with nonuniformly refined meshes finally we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid v cycle algorithms

brief consistency analysis of subspace identification methods based on a linear regression approach in the literature results can be found which claim consistency for the subspace method under certain quite weak assumptions unfortunately a new result gives a counter example showing inconsistency under these assumptions and then gives new more strict sufficient assumptions which however does not include important model structures such as e g box jenkins based on a simple least squares approach this paper shows the possible inconsistency under the weak assumptions and develops only slightly stricter assumptions sufficient for consistency and which includes any model structure

subspace based methods for the identification of linear time invariant systems subspace based methods for system identification have attracted much attention during the past few years this interest is due to the ability of providing accurate state space models for multivariable linear systems directly from input output data the methods have their origin in classical state space realization theory as developed in the 1960s the main computational tools are the qr and the singular value decompositions here an overview of existing subspace based techniques for system identification is given the methods are grouped into the classes of realization based and direct techniques similarities between different algorithms are pointed out and their applicability is commented upon we also discuss some recent ideas for improving and extending the methods a simulation example is included for comparing different algorithms the subspace based approach is found to perform competitive with respect to prediction error methods provided the system is properly excited

nonhomogeneous neumann problem for the poisson equation in domains with an external cusp abstract in this work we analyze the existence and regularity of the solution of a nonhomogeneous neumann problem for the poisson equation in a plane domain with an external cusp in order to prove that there exists a unique solution in h 1 using the lax milgram theorem we need to apply a trace theorem since is not a lipschitz domain the standard trace theorem for h 1 does not apply in fact the restriction of h 1 functions is not necessarily in l 2 so we introduce a trace theorem by using weighted sobolev norms in under appropriate assumptions we prove that the solution of our problem is in h 2 and we obtain an a priori estimate for the second derivatives of the solution

a novel subspace identification approach with enforced causal models subspace identification methods sims for estimating state space models have been proven to be very useful and numerically efficient they exist in several variants but all have one feature in common as a first step a collection of high order arx models are estimated from vectorized input output data in order not to obtain biased estimates this step must include future outputs however all but one of the submodels include non causal input terms the coefficients of them will be correctly estimated to zero as more data become available they still include extra model parameters which give unnecessarily high variance and also cause bias for closed loop data in this paper a new model formulation is suggested that circumvents the problem within the framework the system matrices a b c d and markov parameters can be estimated separately it is demonstrated through analysis that the new methods generally give smaller variance in the estimate of the observability matrix and it is supported by simulation studies that this gives lower variance also of the system invariants such as the poles

analysis of a robust finite element approximation for a parabolic equation with rough boundary data the approximation of parabolic equations with nonhomogeneous dirichlet boundary data by a numerical method that consists of finite elements for the space discretization and the backward euler time discretization is studied the boundary values are assumed in a least squares sense it is shown that this method achieves an optimal rate of convergence for rough only l2 boundary data and for smooth data as well the results of numerical computations which confirm the robust theoretical error estimates are also presented

adaptive finite elements for exterior domain problems we present an adaptive finite element method for solving elliptic problems in exterior domains that is for problems in the exterior of a bounded closed domain in bbb r d d in 2 3 we describe a procedure to generate a sequence of bounded computational domains omega h k k 1 2 more precisely a sequence of successively finer and larger grids until the desired accuracy of the solution u h is reached to this end we prove an a posteriori error estimate for the error on the unbounded domain in the energy norm by means of a residual based error estimator furthermore we prove convergence of the adaptive algorithm numerical examples show the optimal order of convergence

finite element approximation of dirichlet boundary control for elliptic pdes on two and three dimensional curved domains we consider the variational discretization of elliptic dirichlet optimal control problems with constraints on the control the underlying state equation which is considered on smooth two and three dimensional domains is discretized by linear finite elements taking into account domain approximation the control variable is not discretized we obtain optimal error bounds for the optimal control in two and three space dimensions and prove a superconvergence result in two dimensions provided that the underlying mesh satisfies some additional condition we confirm our analytical findings by numerical experiments

approximations of elliptic optimal control problems with controls acting on a lower dimensional manifold in this paper we study finite element approximations to some elliptic optimal control problems with controls acting on a lower dimensional manifold which can be a point a curve or a surface we use piecewise linear finite elements to approximate state variables while utilizing the variational discretization to approximate control variables we derive several a priori error estimates for optimal controls from different cases depending on the dimensions of the computational domain and the manifold where controls act we ends up with extensive numerical experiments which confirm our theoretical findings