the convergence of spline collocation for strongly elliptic equations on curves most boundary element methods for two dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order the equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines the crucial assumption for the generalized boundary integral and integro differential operators is strong ellipticity our analysis is based on simple fourier expansions in particular we extend results by j saranen and w l wendland from constant to variable coefficient equations our results include the first convergence proof of midpoint collocation with piecewise constant functions i e the panel method for solving systems of cauchy singular integral equations

a bie method for a nonlinear bvp abstract a nonlinear boundary value problem bvp governed by laplace s equation with a nonlinear boundary condition is considered this paper is concerned with the numerical solvability of a nonlinear boundary integral equation bie obtained by reformulating the nonlinear bvp we give a simple alternative to the standard collocation method for the nonlinear bie which requires less computational effort the method is a bie analogue of the product approximation method used for solving some nonlinear problems by the finite element method we obtain superconvergence of approximate solutions introduced in this paper

an adaptive finite element moreau yosida based solver for a coupled cahn hilliard navier stokes system an adaptive a posteriori error estimator based finite element method for the numerical solution of a coupled cahn hilliard navier stokes system with a double obstacle homogenous free interfacial energy density is proposed a semi implicit euler scheme for the time integration is applied which results in a system coupling a quasi stokes or oseen type problem for the fluid flow to a variational inequality for the concentration and the chemical potential according to the cahn hilliard model 16 a moreau yosida regularization is employed which relaxes the constraints contained in the variational inequality and thus enables semi smooth newton solvers with locally superlinear convergence in function space moreover upon discretization this yields a mesh independent method for a fixed relaxation parameter for the finite dimensional approximation of the concentration and the chemical potential piecewise linear and globally continuous finite elements are used and for the numerical approximation of the fluid velocity taylor hood finite elements are employed the paper ends by a report on numerical examples showing the efficiency of the new method

on nonsymmetric saddle point matrices that allow conjugate gradient iterations linear systems in saddle point form are usually highly indefinite which often slows down iterative solvers such as krylov subspace methods it has been noted by several authors that negating the second block row of a symmetric indefinite saddle point matrix leads to a nonsymmetric matrix mathcal a whose spectrum is entirely contained in the right half plane in this paper we study conditions so that mathcal a is diagonalizable with a real and positive spectrum these conditions are based on necessary and sufficient conditions for positive definiteness of a certain bilinear form with respect to which mathcal a is symmetric in case the latter conditions are satisfied there exists a well defined conjugate gradient cg method for solving linear systems with mathcal a we give an efficient implementation of this method discuss practical issues such as error bounds and present numerical experiments

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

a revised modified cholesky factorization algorithm a modified cholesky factorization algorithm introduced originally by gill and murray and refined by gill murray and wright is used extensively in optimization algorithms since its introduction in 1990 a different modified cholesky factorization of schnabel and eskow has also gained widespread usage compared with the gill murray wright algorithm the schnabel eskow algorithm has a smaller a priori bound on the perturbation added to ensure positive definiteness and some computational advantages especially for large problems users of the schnabel eskow algorithm however have reported cases from two different contexts where it makes a far larger modification to the original matrix than is necessary and than is made by the gill murray wright method this paper reports on a simple modification to the schnabel eskow algorithm that appears to correct all the known computational difficulties with the method without harming its theoretical properties or its computational behavior in any other cases in new computational tests the modifications to the original matrix made by the new algorithm appear virtually always to be smaller than those made by the gill murray wright algorithm sometimes by significant amounts the perturbed matrix is allowed to be more ill conditioned with the new algorithm but this seems to be appropriate in the known contexts where the underlying problem is ill conditioned

the convergence of even degree spline collocation solution for potential problems in smooth domains of the plane in this article we derive new error estimates for collocation solution of potential type problems by using even degree smooth splines as trial functions it turns out that for smooth potentials the assured convergence is of the same order as by using splines of the odd degreed 1 some numerical examples which conform the theoretical results are presented

on the boundary element method for some nonlinear boundary value problems here we analyse the boundary element galerkin method for two dimensional nonlinear boundary value problems governed by the laplacian in an interior or exterior domain and by highly nonlinear boundary conditions the underlying boundary integral operator here can be decomposed into the sum of a monotoneous hammerstein operator and a compact mapping we show stability and convergence by using leray schauder fixed point arguments due to petryshyn and necas

block triangular preconditioners for saddle point problems with a penalty term block triangular preconditioners for a class of saddle point problems with a penalty term are considered an important example is the mixed formulation of the pure displacement problem in linear elasticity it is shown that the spectrum of the preconditioned system is contained in a real positive interval and that the interval bounds can be made independent of the discretization and penalty parameters this fact is used to construct bounds of the convergence rate of the gmres method with respect to an energy norm numerical results are given for gmres and bi cgstab

projection methods for a class of hammerstein equations here some new convergence results for a class of hammerstein equations are given a particular example appearing in the solution of a steady state heat conduction problem with a nonlinear boundary condition was recently studied by ruotsalainen and wendland numer math 53 1988 pp 299 314 using the galerkin approximation and by ruotsalainen and saranen jcam 28 1989 pp 339 348 using collocation this paper improves the convergence results in both of these works the general result also applies to some time dependent nonlinear model problems

analysis of the minimal residual method applied to ill posed optimality systems we analyze the performance of the minimal residual minres method applied to linear karush kuhn tucker systems arising in connection with inverse problems such optimality systems typically have a saddle point structure and have unique solutions for all alpha 0 where alpha is the parameter employed in the tikhonov regularization unfortunately the associated spectral condition number is very large for small values of alpha which strongly indicates that their numerical treatment is difficult our main result shows that a broad range of linear ill posed optimality systems can be solved efficiently with the minres method this result is obtained by carefully analyzing the spectrum of the associated saddle point operator except for a few isolated eigenvalues the spectrum consists of three bounded intervals krylov subspace methods handle such problems very well for severely ill posed cases techniques based on chebyshev polynomials are applied to prove that the number of iterations needed by

a parallel two level domain decomposition based one shot method for shape optimization problems summary a two level domain decomposition method is introduced for general shape optimization problems constrained by the incompressible navier stokes equations the optimization problem is first discretized with a finite element method on an unstructured moving mesh that is implicitly defined without assuming that the computational domain is known and then solved by some one shot lagrange newton krylov schwarz algorithms in this approach the shape of the domain its corresponding finite element mesh the flow fields and their corresponding lagrange multipliers are all obtained computationally in a single solve of a nonlinear system of equations highly scalable parallel algorithms are absolutely necessary to solve such an expensive system the one level domain decomposition method works reasonably well when the number of processors is not large aiming for machines with a large number of processors and robust nonlinear convergence we introduce a two level inexact newton method with a hybrid two level overlapping schwarz preconditioner as applications we consider the shape optimization of a cannula problem and an artery bypass problem in 2d numerical experiments show that our algorithm performs well on a supercomputer with over 1000 processors for problems with millions of unknowns copyright 2014 john wiley sons ltd

modified cholesky algorithms a catalog with new approaches given an n n symmetric possibly indefinite matrix a a modified cholesky algorithm computes a factorization of the positive definite matrix a e where e is a correction matrix since the factorization is often used to compute a newton like downhill search direction for an optimization problem the goals are to compute the modification without much additional cost and to keep a e well conditioned and close to a gill murray and wright introduced a stable algorithm with a bound of e 2 o n 2 an algorithm of schnabel and eskow further guarantees e 2 o n we present variants that also ensure e 2 o n more and sorensen and cheng and higham used the block lbl t factorization with blocks of order 1 or 2 algorithms in this class have a worst case cost o n 3 higher than the standard cholesky factorization we present a new approach using a sandwiched ltl t lbl t factorization with t tridiagonal that guarantees a modification cost of at most o n 2

shishkin mesh simulation a new stabilization technique for convection diffusion problems abstract a new stabilization procedure is presented it is based on a simulation of the interaction between the coarse and fine parts of a shishkin mesh but can be applied on coarse and irregular meshes and on domains with nontrivial geometries the technique which does not require adjusting any parameter can be applied to different stabilized and non stabilized methods numerical experiments show it to obtain oscillation free approximations on problems with boundary and internal layers on uniform and nonuniform meshes and on domains with curved boundaries

on the stability of periodic solutions for differential equations with a fredholm operator at the highest derivative abstract on the basis of a classic floquet theory variant the stability of periodic solutions at andronov hopf bifurcation for autonomous nonlinear differential equations of s th order s 1 with a degenerate fredholm operator at the highest derivative in banach spaces is investigated stability criteria based on the perturbation theory and newton diagram method are suggested

on the principle of reduced stability abstract the principle of reduced stability says that the stability of bifurcating stationary or periodic solutions is given by the finite dimensional bifurcation equation obtained by the method of lyapunov schmidt to be more precise the linearized stability is governed by the linearization of the bifurcation equation about the bifurcating branch of solutions and in particular by the signs of the real parts of the perturbation of the eigenvalues along this branch this principle is true for simple eigenvalue bifurcation whereas it may be false for higher dimensional bifurcation equations a condition for the validity of that principle is given a counterexample shows that it cannot be dropped in general

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

on the convergence of the collocation method for nonlinear boundary integral equations abstract recently galerkin and collocation methods have been analysed for some nonlinear boundary integral equations for the collocation method it has been assumed that the nonlinearity is asymptotically linear in this paper we remove this restriction we shall prove the convergence of the collocation method for nonlinear boundary integral equations when the nonlinearity has a polynomial growth condition in addition to this the optimal order error estimates follow in l q norm

on the collocation method for a nonlinear boundary integral equation abstract in this paper we study a potential problem with a nonlinear boundary condition using the green representation formula for a harmonic function we reformulate the nonlinear boundary value problem as a nonlinear boundary integral equation we shall give a brief discussion of the solvability of the integral equation the aim of this paper however is to analyse the collocation method for finding an approximate solution to this equation using the theory of a proper and a stable mappings we prove the unique solvability of the collocation equations and the asymptotic error estimates to do this we assume that the nonlinearity is strongly monotone

changing the norm in conjugate gradient type algorithms a conjugate gradient type method for solving ax b or more precisely an orthogonal error method operatorname oe b c a is determined by a formal inner product matrix b and a left preconditioning matrix c relations are pointed out between quantities such as iterates residuals direction vectors and recurrence coefficients of orthogonal error methods with different but in a particular way related matrices b namely for operatorname oe b ca k c a k 0 1 ldots and for operatorname oe a h c h k b c a k 0 1 ldots the relations for the first sequence of methods have to do with rutishauser s lr algorithm those for the second one are based on a generalization of the schonauer weiss smoothing algorithm relevant for practice are the cases k 0 1 for b ca and b ca h

operator preconditioning for a class of inequality constrained optimal control problems we propose and analyze two strategies for preconditioning linear operator equations that arise in pde constrained optimal control in the framework of conjugate gradient methods our particular focus is on control or state constrained problems where we consider the question of robustness with respect to critical parameters we construct a preconditioner that yields favorable robustness properties with respect to critical parameters

lp estimates of boundary integral equations for some nonlinear boundary value problems recently galerkin and collocation methods have been analyzed for boundary integral equation formulations of some potential problems in the plane with nonlinear boundary conditions and stability results and error estimates in theh 1 2 norm have been proved ruotsalainen and wendland and ruotsalainen and saranen we show that these results extend tol p setting without any extra conditions these extensions are proved by studying the uniform boundedness of the inverses of the linearized integral operators and then considering the nonlinear equations the fact that inh 1 2 setting the nonlinear operator is a homeomorphism with lipschitz continuous inverse plays a crucial role optimal error estimates for the galerkin and collocation method inl p space then follow

preconditioning saddle point systems with applications in optimization saddle point systems arise in many applications areas in fact in any situation where an extremum principle arises with constraints the stokes problem describing slow viscous flow of an incompressible fluid is a classic example coming from pdes and in the area of optimization such problems are ubiquitous in this paper we present a framework into which many well known methods for solving saddle point systems fit based on this description we show how new approaches for the solution of saddle point systems arising in optimization can be derived from the bramble pasciak conjugate gradient approach widely used in pdes and more recent generalizations thereof in particular we derive a class of new solution methods based on the use of preconditioned conjugate gradients in nonstandard inner products and demonstrate how these can be understood through more standard machinery we show connections to constraint preconditioning and give the results of numerical computations on a number of standard optimization test examples

the numerical solution of a nonlinear hypersingular boundary integral equation in this paper we consider a direct hypersingular integral approach to solve harmonic problems with nonlinear boundary conditions by using a practical variant of the galerkin boundary element method the proposed approach provides an almost optimal balance between the order of convergence and the numerical effort of work to compute the approximate solution numerical examples confirm the theoretical results

on the eigenvalues of a class of saddle point matrices we study spectral properties of a class of block 2 2 matrices that arise in the solution of saddle point problems these matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system we give conditions for having a positive real spectrum and for ensuring diagonalizability of the matrix in particular we show that these properties hold for the discrete stokes operator and we discuss the implications of our characterization for augmented lagrangian formulations for krylov subspace solvers and for certain types of preconditioners

boundary element methods for potential problems with nonlinear boundary conditions galerkin boundary element methods for the solution of novel first kind steklov poincare and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two and three dimensional lipschitz domains with nonlinear boundary conditions for the numerical solution of the resulting newton iterate linear boundary integral equations we propose practical variants of the galerkin scheme and give corresponding error estimates we also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy

a taxonomy for conjugate gradient methods the conjugate gradient method of hestenes and stiefel is an effective method for solving large sparse hermitian positive definite hpd systems of linear equations ax b generalizations to non hpd matrices have long been sought the recent theory of faber and manteuffel gives necessary and sufficient conditions for the existence of a cg method this paper uses these conditions to develop and organize such methods it is shown that any cg method for ax b is characterized by an hpd inner product matrix b and a left preconditioning matrix c at each step the method minimizes the b norm of the error over a krylov subspace this characterization is then used to classify known and new methods finally it is shown how eigenvalue estimates may be obtained from the iteration parameters generalizing the well known connection between cg and lanczos such estimates allow implementation of a stopping criterion based more nearly on the true error

on the convergence of the galerkin method for nonsmooth solutions of integral equations it is shown that the stability region of the galerkin method includes solutions not lying in the conventional energy space optimal order error estimates for these nonsmooth solutions are derived the new result is compared with the classical statement by means of the basic potential problem