a priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1d for a next nearest neighbour pair interaction model in a periodic domain a priori and a posteriori analyses of the quasinonlocal quasicontinuum method qnl qc are presented the results are valid for large deformations and essentially guarantee a one to one correspondence between atomistic solutions and qnl qc solutions the analysis is based on consistency error estimates in negative norms novel a priori and a posteriori stability estimates and a quantitative inverse function theorem

a posteriori and a priori error analysis for finite element approximations of self adjoint elliptic eigenvalue problems we present a new error analysis for finite element approximations of self adjoint elliptic eigenvalue problems the analysis consists of three steps first we prove a posteriori estimates for the error in the approximate eigenvectors and eigenvalues the error in the eigenvectors is measured both in the l2 and energy norms in these estimates the error is bounded in terms of the mesh size a stability factor and the residual obtained by inserting the approximate eigenpair into the differential equation the stability factor describes relevant stability properties of the continuous problem and we give a precise estimate of its size in terms of the spectrum of the continuous problem the mesh size and the choice of norm next we prove an a priori estimate of the residual in terms of derivatives of the exact eigenvectors and the mesh size finally we obtain precise a priori error estimates by combination of the a posteriori error estimates with the a priori residual estimate the analysis shows that the a posteriori estimates are optimal and may be used for quantitative error estimation and design of adaptive algorithms

a posteriori error analysis for time dependent ginzburg landau type equations this work presents an a posteriori error analysis for the finite element approximation of time dependent ginzburg landau type equations in two and three space dimensions the solution of an elliptic self adjoint eigenvalue problem as a post processing procedure in each time step of a finite element simulation leads to a fully computable upper bound for the error theoretical results for the stability of degree one vortices in ginzburg landau equations and of generic interfaces in allen cahn equations indicate that the error estimate only depends on the inverse of a small parameter in a low order polynomial the actual dependence of the error estimate upon this parameter is explicitly determined by the computed eigenvalues and can therefore be monitored within an approximation scheme the error bound allows for the introduction of local refinement indicators which may be used for adaptive mesh and time step size refinement and coarsening numerical experiments underline the reliability of this approach

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

equilibrated residual error estimator for edge elements reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem a cheap computation of such functions via equilibration is well known for scalar equations of second order we simplify and modify the equilibration such that it can be applied to the curl curl equation and edge elements the construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions for this reason raviart thomas elements are extended in the spirit of distributions

linear convergence of an adaptive finite element method for the p laplacian equation we study an adaptive finite element method for the p laplacian like pdes using piecewise linear continuous functions the error is measured by means of the quasi norm of barrett and liu we provide residual based error estimators without a gap between the upper and lower bound we show linear convergence of the algorithm which is similar to the one of morin nochetto and siebert all results are obtained without extra marking for the oscillation

justification of the cauchy born approximation of elastodynamics we present sharp convergence results for the cauchy born approximation of general classical atomistic interactions for static problems with small data and for dynamic problems on a macroscopic time interval

numerical verifications of solutions for elliptic equations in nonconvex polygonal domains in this paper methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied in order to verify solutions using computer it is necessary to determine some constants which appear in a priori error estimations we propose some methods for determination of these constants in numerical examples calculating these constants for anl shaped domain we verify the solution of a nonlinear elliptic equation

computer assisted enclosure methods for elliptic differential equations abstract we report on methods for computing enclosures of solutions of second order nonlinear elliptic boundary value problems simultaneously proving the existence of a solution in the enclosing set the old fashioned monotonicity methods are well suited for this task but only for a restricted class of problems therefore we propose a new approach which is based on a suitable fixed point formulation of the problem and uses as an essential ingredient norm bounds for the inverse of the linearization of the given problem at some approximate solution which is computed numerically these norm bounds are obtained via eigenvalue enclosures we also give a brief description of an alternative method proposed by m t nakao

a posteriori existence in numerical computations a typical a posteriori error analysis for numerical methods for differential equations consists of two elements the stability of the continuous problem and an a posteriori computed residual estimate if the problem is nonlinear then this is possible only if it is strongly monotone in this paper we will estimate the stability constant a posteriori as well in order to apply the inverse function theorem to derive the existence of solutions from the numerical computation and thus obtain rigorous a posteriori error estimates for nonmonotone equations

an optimal adaptive finite element method although existing adaptive finite element methods for solving second order elliptic equations often perform better in practical computations than nonadaptive ones usually they are not even proven to converge only recently in the work of dorfler siam j numer anal 33 1996 pp 1106 1124 and that of morin nochetto and siebert siam j numer anal 38 2000 pp 466 488 adaptive methods were constructed for which convergence could be demonstrated however convergence alone does not imply that the method is more efficient than its nonadaptive counterpart in numer math 97 2004 pp 219 268 binev dahmen and devore added a coarsening step to the routine of morin nochetto and siebert and proved that the resulting method is quasi optimal in the following sense if the solution is such that for some s 0 the error in energy norm of the best continuous piecewise linear approximations subordinate to any partition with n triangles is mathcal o n s then given an eps 0 the adaptive method produces an approximation with an error less than eps subordinate to a partition with mathcal o eps 1 s triangles in only mathcal o eps 1 s operations in this paper employing a different type of adaptive partition we develop an adaptive method with properties similar to those of binev dahmen and devore s method but unlike their method our coarsening routine will be based on a transformation to a wavelet basis and we expect it to have better quantitative properties furthermore all our results are valid uniformly in the size of possible jumps of the diffusion coefficients since the algorithm uses solely approximations of the right hand side we can even allow right hand sides in h 1 omega that lie outside l 2 omega at least when they can be sufficiently well approximated by piecewise constants in our final adaptive algorithm all tolerances depend on an a posteriori estimate of the current error instead of an a priori one this can be expected to provide quantitative advantages

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

the qr algorithm for unitary hessenberg matrices abstract let h be an n n unitary right hessenberg matrix with positive subdiagonal elements using what we call the schur parameterization of h we show how one step of the shifted qr algorithm for h can be carried out in o n arithmetic operations coupled with the shift strategy of eberlein and huang 3 this will permit computation of the spectrum of h to machine precision in o n 2 operations one potential application is the computation of gauss szego quadrature formulas 12 given the associated schur parameters 7 the weights can also be computed by direct analogy with 6

eigenvalue computation for unitary rank structured matrices in this paper we describe how to compute the eigenvalues of a unitary rank structured matrix in two steps first we perform a reduction of the given matrix into hessenberg form next we compute the eigenvalues of this resulting hessenberg matrix via an implicit qr algorithm along the way we explain how the knowledge of a certain shift correction term to the structure can be used to speed up the qr algorithm for unitary hessenberg matrices and how this observation was implicitly used in a paper due to william b gragg we also treat an analogue of this observation in the hermitian tridiagonal case

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

validated continuation for equilibria of pdes one of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor corrector continuation techniques in the case of partial differential equations this procedure must be applied to some finite dimensional approximation which of course raises the question of the validity of the output we introduce a new technique that combines the information obtained from the predictor corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite dimensional partial differential equation using the cahn hilliard and swift hohenberg equations as models we demonstrate that the cost of this new validated continuation is less than twice the cost of the standard continuation method alone

six line configurations and string dualities we study the family of k3 surfaces of picard rank sixteen associated with the double cover of the projective plane branched along the union of six lines and the family of its van geemen sarti partners i e k3 surfaces with special nikulin involutions such that quotienting by the involution and blowing up recovers the former we prove that the family of van geemen sarti partners is a four parameter family of k3 surfaces with h oplus e 7 1 oplus e 7 1 lattice polarization we describe explicit weierstrass models on both families using even modular forms on the bounded symmetric domain of type iv we also show that our construction provides a geometric interpretation called geometric two isogeny for the f theory heterotic string duality in eight dimensions

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

consistency stability a priori and a posteriori errors for petrov galerkin methods applied to nonlinear problems in an abstract framework we present a formalism which specifies the notions of consistency and stability of petrov galerkin methods used to approximate nonlinear problems which are in many practical situations strongly nonlinear elliptic problems this formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems this theory is illustrated with the example div k u u c u f in a two dimensional domain with dirichlet boundary conditions

finite element approximations of nonlinear elastic waves in this paper we study finite element methods for a class of problems of nonlinear elastodynamics we discretize the equations in space using galerkin methods for the temporal discretization the construction of our schemes is based on rational approximations of cos x and ex we analyze semidiscrete as well as secondand fourth order accurate in time fully discrete methods for the approximation of the solution of the problem and prove optimal order l2 error estimates for some schemes a taylor type technique is used so that only linear systems of equations need be solved at each time step in the proofs we need various estimates for a nonlinear elliptic projection the proofs of which are also established in the paper

force based atomistic continuum blending for multilattices we formulate the blended force based quasicontinuum method for multilattices and develop rigorous error estimates in terms of the approximation parameters choice of atomistic region blending region and continuum finite element mesh balancing the approximation parameters yields a convergent atomistic continuum multiscale method for multilattices with point defects including a rigorous convergence rate in terms of the computational cost the analysis is illustrated with numerical results for a stone wales defect in graphene