a homographic best approximation problem with application to optimized schwarz waveform relaxation we present and study a homographic best approximation problem which arises in the analysis of waveform relaxation algorithms with optimized transmission conditions its solution characterizes in each class of transmission conditions the one with the best performance of the associated waveform relaxation algorithm we present the particular class of first order transmission conditions in detail and show that the new waveform relaxation algorithms are well posed and converge much faster than the classical one the number of iterations to reach a certain accuracy can be orders of magnitudes smaller we illustrate our analysis with numerical experiments

spectral hp least squares finite element formulation for the navier stokes equations we consider the application of least squares finite element models combined with spectral hp methods for the numerical solution of viscous flow problems the paper presents the formulation validation and application of a spectral hp algorithm to the numerical solution of the navier stokes equations governing two and three dimensional stationary incompressible and low speed compressible flows the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity or velocity gradients as additional independent variables and the least squares method is used to develop the finite element model high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method spectral convergence of the l2 least squares functional and l2 error norms is verified using smooth solutions to the two dimensional stationary poisson and incompressible navier stokes equations numerical results for flow over a backward facing step steady flow past a circular cylinder three dimensional lid driven cavity flow and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation

quasi optimized schwarz methods for reaction diffusion equations with time delay abstract in this paper we investigate the convergence behavior of the schwarz waveform relaxation swr algorithms for solving pdes with time delay we choose the reaction diffusion equations with a constant time delay as the underlying model problem and try to derive optimized transmission conditions of robin type to this end we propose a new method to get quasi optimized parameter involved in the transmission conditions and it is shown that this method is essentially different from the existing ones moreover when the situation is reduced into the heat equations with a constant delay we show that this method results in a more efficient quasi optimized parameter numerical results are provided to validate our theoretical results

space time continuous analysis of waveform relaxation for the heat equation waveform relaxation algorithms for partial differential equations pdes are traditionally obtained by discretizing the pde in space and then splitting the discrete operator using matrix splittings for the semidiscrete heat equation one can show linear convergence on unbounded time intervals and superlinear convergence on bounded time intervals by this approach however the bounds depend in general on the mesh parameter and convergence rates deteriorate as one refines the mesh motivated by the original development of waveform relaxation in circuit simulation where the circuits are split in the physical domain into subcircuits we split the pde by using overlapping domain decomposition we prove linear convergence of the algorithm in the continuous case on an infinite time interval at a rate depending on the size of the overlap this result remains valid after discretization in space and the convergence rates are robust with respect to mesh refinement the algorithm is in the class of waveform relaxation algorithms based on overlapping multisplittings our analysis quantifies the empirical observation by jeltsch and pohl siam j sci comput 16 1995 pp 40 49 that the convergence rate of a multisplitting algorithm depends on the overlap numerical results are presented which support the convergence theory

overlapping schwarz waveform relaxation method for the solution of the reaction diffusion equation abstract we are interested in solving time dependent problems using domain decomposition methods in the classical approach one discretizes first the time dimension and then one solves a sequence of steady problems by a domain decomposition method in this article we treat directly the time dependent problem and we study a schwarz waveform relaxation algorithm for the convection diffusion equation we study the convergence of the overlapping schwarz waveform relaxation method for solving the reaction diffusion equation over multi overlapped subdomains also we will show that the method converges linearly and superlinearly over long and short time intervals and the convergence depends on the size of overlap numerical results are presented from solutions of a specific model problems to demonstrate the convergence linear and superlinear and the role of the overlap size

error analysis of some finite element methods for the stokes problem we prove the optimal order of convergence for some two dimensional finite element methods for the stokes equations first we consider methods of the taylor hood type the triangular p3 p2 element and the qk qk 1 i k 2 family of quadrilateral elements then we introduce two new low order methods with piecewise constant approximations for the pressure the analysis is performed using our macroelement technique which is reviewed in a slightly altered form

on monotone iteration and schwarz methods for nonlinear parabolic pdes the schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains the solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains in this paper proofs of convergence of some schwarz alternating methods for nonlinear parabolic problems which are known to have solutions by the monotone method also known as the method of subsolutions and supersolutions are given in particular an additive schwarz method for scalar as well some coupled nonlinear pdes are shown to converge to the solution on finitely many subdomains in the coupled system case each subdomain pde is linear decoupled and can be solved concurrently with other subdomain pdes these results are applicable to several models in population biology the convergence behavior is illustrated by two numerical examples

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

optimal schwarz waveform relaxation for the one dimensional wave equation we introduce a nonoverlapping variant of the schwarz waveform relaxation algorithm for wave propagation problems with variable coefficients in one spatial dimension we derive transmission conditions which lead to convergence of the algorithm in a number of iterations equal to the number of subdomains independently of the length of the time interval these optimal transmission conditions are in general nonlocal but we show that the nonlocality depends on the time interval under consideration and we introduce time windows to obtain optimal performance of the algorithm with local transmission conditions in the case of piecewise constant wave speed we show that convergence in two iterations can be achieved independently of the number of subdomains in that case the algorithm thus scales optimally with the number of subdomains provided the time windows are chosen appropriately for continuously varying coefficients we prove convergence of the algorithm with local transmission conditions using energy estimates we then introduce a finite volume discretization which permits computations on nonmatching grids and we prove convergence of the fully discrete schwarz waveform relaxation algorithm we finally illustrate our analysis with numerical experiments

analysis of the parallel schwarz method for growing chains of fixed sized subdomains part i in implicit solvation models the electrostatic contribution to the solvation energy can be estimated by solving a system of elliptic partial differential equations modeling the reaction potential the domain of definition of such elliptic equations is the union of the van der waals cavities corresponding to the atoms of the solute molecule therefore the computations can naturally be performed using schwarz methods where each atom of the molecule corresponds to a subdomain in contrast to classical schwarz theory it was observed numerically that the convergence of the schwarz method in this case does not depend on the number of subdomains even without coarse correction we prove this observation by analyzing the schwarz iteration matrices in fourier space and evaluating corresponding norms in a simplified setting in order to obtain our contraction results we had to choose a specific iteration formulation and we show that other formulations of the same algorithm can generate schwarz iteration matri

energy estimates for continuous and discretized electro reaction diffusion systems abstract we consider electro reaction diffusion systems consisting of continuity equations for a finite number of species coupled with a poisson equation we take into account heterostructures anisotropic materials and rather general statistic relations we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly the same properties are shown for an implicit time discretized version of the problem moreover we provide a space discretized scheme for the electro reaction diffusion system which is dissipative the free energy decays monotonously on a fixed grid we use for each species different voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species

optimization of transmission conditions in waveform relaxation techniques for rc circuits waveform relaxation techniques have become increasingly important with the wide availability of parallel computers with a large number of processors a limiting factor for classical waveform relaxation however is the convergence speed for an important class of problems especially if long time windows are considered in contrast the optimized waveform relaxation algorithm discussed in this paper is well suited to address this problem today several numerical analyses have shown that optimized waveform relaxation algorithms can overcome slow convergence over long time windows however the optimized waveform relaxation techniques require the determination of optimized parameters in this paper we present a theoretical foundation for the determination of the optimized parameters for an important class of rc circuits

optimized schwarz waveform relaxation for advection reaction diffusion equations in two dimensions optimized schwarz waveform relaxation methods have been developed over the last decade for the parallel solution of evolution problems they are based on a decomposition in space and an iteration where only subproblems in space time need to be solved each subproblem can be simulated using an adapted numerical method for example with local time stepping or one can even use a different model in different subdomains which makes these methods very suitable also from a modeling point of view for rapid convergence however it is important to use effective transmission conditions between the space time subdomains and for best performance these transmission conditions need to take the physics of the underlying evolution problem into account the optimization of these transmission conditions leads to mathematically hard best approximation problems of homographic functions we study in this paper in detail the best approximation problem for the case of linear advection reaction diffusion equations in two spatial dimensions we prove comprehensively best approximation results for transmission conditions of robin and ventcel higher order type which can also be used in the various limits for example for the heat equation since we include in our analysis a positive low frequency limiter both in space and time we give for each case closed form asymptotic values for the parameters which can directly be used in implementations of these algorithms and which guarantee asymptotically best performance of the iterative methods we finally show extensive numerical experiments including cases not covered by our analysis for example decompositions with cross points we use q1 finite element discretizations in space and forward and backward euler discretizations in time other discretization could also have been considered since all our analysis is at the continuous level and in all cases we measure performance corresponding to our analysis

a schwarz waveform moving mesh method an r refinement moving mesh method is considered for solving time dependent partial differential equations pdes the resulting coupled system consisting of the physical pde and a moving mesh pde is solved by a schwarz waveform relaxation method in particular the computational space time domain is decomposed into overlapping subdomains and the solution obtained by iteratively solving the system of pdes on each subdomain dirichlet boundary conditions are used to pass solution information between neighboring regions the efficacy of this approach is demonstrated for some model problems for problems where the solutions evolve on disparate time scales in different regions of the spatial domain this approach demonstrates the significant savings in computational time and effort which are possible

resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations we consider a general method for the macroscale modelling of nonlinear nonautomomous partial differential equations or difference equations our method is able to resolve the complex fluctuations which arise in nonliner nonautonomous systems these fluctuations are not accounted for by popular multiscale averaging procedures and homogenization our method is illustrated with the example of a forced burgers equation coarse grained macroscale spatial discretisations of nonlinear nonautonomous partial differential difference equations are given novel support by centre manifold theory dividing the physical domain into overlapping macroscale elements empowers the approach to resolve significant subgrid microscale structures and interactions between neighbouring elements the crucial aspect of this approach is that centre manifold theory organises the resolution of the detailed subgrid microscale structure interacting via the nonlinear dynamics within and between neighbouring elements the techniques and theory developed here may be applied to soundly discretise on a macroscale many dissipative nonautonomous partial differential difference equations such as the forced burgers equation adopted here as an illustrative example

a family of higher order mixed finite element methods for plane elasticity the dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the raviart thomas mixed finite elements for a scalar second order elliptic equation estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inl 2 and optimal order negative norm estimates are obtained inh s for a range ofs depending on the index of the finite element space an optimal order estimate inl for the displacement error is given also a quasioptimal estimate is derived in an appropriate space all estimates are valid uniformly with respect to the compressibility and apply in the incompressible case the formulation of the elements is presented in detail

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

convergence analysis of two level space time additive schwarz method for parabolic equations the additive schwarz as method was originally designed for preconditioning elliptic problems and recently the method was extended successfully as a coupled space time preconditioner for parabolic problems however the existing theory for the additive schwarz method doesn t apply directly to the space time discretized problems in this paper we develop an optimal convergence theory for the two level space time additive schwarz preconditioner we obtain lower and upper bounds for the spectrum of the as preconditioned operator and their dependency on the fine coarse mesh sizes in space and time the number of subdomains and the window size i e the number of coupled time steps some numerical experiments are reported to confirm the theory

absorbing boundary conditions for the wave equation and parallel computing absorbing boundary conditions have been developed for truncating unbounded computational domains to bounded ones the authors investigate the use of absorbing boundary conditions in domain decomposition algorithms for the wave equation moreover they optimize the absorbing boundary conditions in order to obtain a good performance of the schwarz waveform relaxation algorithm the analysis given is confirmed by numerical experiments

on the construction of optimal mixed finite element methods for the linear elasticity problem the mixed finite element method for the linear elasticity problem is considered we propose a systematic way of designing methods with optimal convergence rates for both the stress tensor and the displacement the ideas are applied in some examples

domain decomposition approaches for mesh generation via the equidistribution principle moving mesh methods based on the equidistribution principle are powerful techniques for the space time adaptive solution of evolution problems solving the resulting coupled system of equations namely the original pde and the mesh pde however is challenging in parallel we propose in this paper several schwarz domain decomposition algorithms for this task we then study in detail the convergence properties of these algorithms applied to the nonlinear mesh pde in one spatial dimension we prove convergence for classical transmission conditions and optimal and optimized variants for the generation of steady equidistributing grids a classical parallel schwarz algorithm is presented and analyzed for the generation of time dependent moving equidistributing grids we conclude our study with numerical experiments

space time domain decomposition for parabolic problems we analyze a space time domain decomposition iteration for a model advection diffusion equation in one and two dimensions the discretization of this iteration is the block red black variant of the waveform relaxation method and our analysis provides new convergence results for this scheme the asymptotic convergence rate is super linear and it is governed by the diffusion of the error across the overlap between subdomains hence it depends on both the size of this overlap and the diffusion coefficient in the equation however it is independent of the number of subdomains provided the size of the overlap remains fixed the convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases the duration of the transient linear regime is proportional to the length of the time window for advection dominated problems the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases moreover it is independent of the size of the time window and of the number of subdomains numerical calculations illustrate our analysis

analysis and finite element approximation of compressible and incompressible linear isotropic elasticity based upon a variational principle abstract we propose a variational principle for compressible and incompressible linear isotropic elasticity involving four dependent variables the strain tensor the augmented stress tensor pressure and displacement the variational formulation derived from the principle is analyzed employing a general result due to brezzi finite element methods using the galerkin and sbb methods are shown to converge for a wide family of finite element interpolations

a new family of stable elements for nearly incompressible elasticity based on a mixed petrov galerkin finite element formulation adding to the classical hellinger reissner formulation another residual form of the equilibrium equation a new petrov galerkin finite element method is derived it fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations including equal order discontinuous stress and continuous displacement interpolations which are unstable within the galerkin approach error estimates are presented using the babuska brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method

a direct time parallel solver by diagonalization for the wave equation with the advent of very large scale parallel computers it has become more and more important to also use the time direction for parallelization when solving evolution problems while there are man

schwarz waveform relaxation algorithms for the linear viscous equatorial shallow water equations we are interested in numerically solving the viscous shallow water equations with a small coriolis force on a large domain specifically we develop and analyze schwarz waveform relaxation algorithms we split the domain of computation into two subdomains and with appropriate transmission conditions an iterative procedure leads to the global solution we first analyze the dirichlet type transmission conditions computing the convergence rate in fourier laplace variables and proving it is less than 1 yields the convergence of the algorithm the algorithm requires an overlap between subdomains and the convergence is slow we propose a better algorithm for which the overlap is unnecessary the transmission conditions are now an approximation of the absorbing boundary conditions we also prove convergence if the domains overlap the arguments are similar to the dirichlet type problem if not we use variational arguments a numerical scheme is then proposed and numerical results are shown which highlight the

aw1 p estimate for solutions to mixed boundary value problems for second order elliptic differential equations on demontre que sous des hypotheses plutot faibles une solution d un probleme aux valeurs limites mixte pour une equation elliptique d ordre 2 est dans l espace de sobolev w 1 p pour p 2

schwarz waveform relaxation with adaptive pipelining schwarz waveform relaxation swr methods have been developed to solve a wide range of diffusion dominated and reaction dominated equations the appeal of these methods stems primarily from their ability to use nonconforming space time discretizations swr methods are consequently well adapted for coupling models with highly varying spatial and time scales the efficacy of swr methods is questionable however since in each iteration one propagates an error across the entire time interval in this manuscript we introduce an adaptive pipeline approach wherein one subdivides the computational domain into space time blocks and adaptively selects the waveform iterates which should be updated given a fixed number of computational workers our method is complementary to existing space and time parallel methods and can be used to obtain additional speedup when the saturation point is reached for other types of parallelism we analyze these waveform relaxation with adaptive pipelining wrap methods to show c

optimized schwarz methods optimized schwarz methods are a new class of schwarz methods with greatly enhanced convergence properties they converge uniformly faster than classical schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical schwarz methods if the overlap is of the order of the mesh parameter which is often the case in practical applications they achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem we analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new finite element tearing and interconnect feti variants

adiabatic filon type methods for highly oscillatory second order ordinary differential equations in this paper we study efficient numerical integrators for linear and nonlinear systems of highly oscillatory second order ordinary differential equations the systems are reformulated as a first order system which is then transformed to adiabatic variables the solution of the transformed system is a smoother function which is more accessible to numerical approximation than the original system we develop filon type methods for linear systems by approximating the integral as a linear combination of function values and derivatives we then present a special combination of filon type methods and waveform relaxation methods for nonlinear systems both types of methods can be used with far larger step sizes than those required by traditional schemes and their performance drastically improves as frequency grows as are illustrated by numerical experiments

overlapping schwarz waveform relaxation method for the solution of the forward backward heat equation in this article we analyzed the convergence of the schwarz waveform relaxation method for solving the forward backward heat equation numerical results are presented for a specific type of model problem

well posedness of the fundamental boundary value problems for constrained anisotropic elastic materials we consider the equations of linear homogeneous anisotropic elasticity admitting the possibility that the material is internally constrained and formulate a simple necessary and sufficient condition for the fundamental boundary value problems to be well posed for materials fulfilling the condition we establish continuous dependence of the displacement and stress on the elastic moduli and ellipticity of the elasticity system as an application we determine the orthotropic materials for which the fundamental problems are well posed in terms of their young s moduli shear moduli and poisson ratios finally we derive a reformulation of the elasticity system that is valid for both constrained and unconstrained materials and involves only one scalar unknown in addition to the displacements for a two dimensional constrained material a further reduction to a single scalar equation is outlined

waveform relaxation with overlapping splittings in this paper an extension of the waveform relaxation algorithm for solving large systems of ordinary differential equations is presented the waveform relaxation algorithm is well suited for parallel computation because it decomposes the solution space into several disjoint subspaces allowing the subspaces to overlap i e dropping the assumption of disjointedness an extension of this algorithm is obtained this new algorithm the so called multisplitting algorithm is also well suited for parallel computation as numerical examples demonstrate this overlapping of the subsystems heavily reduces the computation time

on time domain simulation of lossless transmission lines with nonlinear terminations a time domain approach is presented to solve nonlinear circuits with lossless transmission lines mathematically the circuits are described by a special kind of nonlinear differential algebraic equations daes with multiple constant delays in order to directly compute these delay systems in time domain decoupling by waveform relaxation wr is applied to the systems for the relaxation based method we provide a new convergence proof numerical experiments are given to illustrate the novel approach