block monotone iterative methods for numerical solutions of nonlinear elliptic equations two block monotone iterative schemes for a nonlinear algebraic system which is a finite difference approximation of a nonlinear elliptic boundary value problem are presented and are shown to converge monotonically either from above or from below to a solution of the system this monotone convergence result yields a computational algorithm for numerical solutions as well as an existence comparison theorem of the system including a sufficient condition for the uniqueness of the solution an advantage of the block iterative schemes is that the thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one dimensional problems the block iterative schemes are compared with the point monotone iterative schemes of picard jacobi and gauss seidel and various theoretical comparison results among these monotone iterative schemes are given these comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given

domain decomposition algorithms for mixed methods for second order elliptic problems domain decomposition algorithms for mixed finite element methods for linear second order elliptic problems in r 2 and r 3 are developed a convergence theory for two level and multilevel schwarz methods applied to the algorithms under consideration is given it is shown that the condition number of these iterative methods is bounded uniformly from above in the same manner as in the theory of domain decomposition methods for conforming and nonconforming finite element methods for the same differential problems numerical experiments are presented to illustrate the present techniques

a two level schwarz method for a finite element approximation of a nonlinear biharmonic equation in this paper we consider a finite element approximation of a nonlinear biharmonic equation which is related to the well known two dimensional navier stokes equations first optimal energy and h 1 norm estimates are obtained second a two level additive schwarz method is presented for the discrete nonlinear algebraic system it is shown that if the reynolds number is sufficiently small the two level schwarz method is optimal i e the convergence rate of the schwarz method is independent of the mesh size and the number of subdomains

geometrically exact sandwich shells the static case the present formulation offers a general method for analyzing the static response of geometrically exact sandwich shells undergoing large deformation the layer directors at a point in the reference surface are connected to each other by universal joints and form a chain of rigid links finite rotations of the directors in every layer are allowed with shear deformation independently accounted for in each layer the thickness and the length of each layer can be arbitrary thus allowing the modeling of an important class of multilayer structures having ply drop offs the present formulation is thus suitable to model shell structures with patches of constrained viscoelastic materials or of piezoelectric materials the nonlinear weak form of the governing equations of equilibrium is constructed here and then the linearization of the weak form and the associated inextensible directors update are derived leading to a symmetric tangent stiffness matrix a galerkin finite element projection of the linearized equilibrium equations results in a system of nonlinear algebraic equations in which the interpolation accounts for the finite rotations of the directors we present extensive numerical examples including sandwich shells with three identical layers and ply drop offs to illustrate the applicability and versatility of the proposed formulation

the schwarz alternating method in solid mechanics abstract we advance the schwarz alternating method as a means for concurrent multiscale coupling in finite deformation solid mechanics we prove that the schwarz alternating method converges to the solution of the problem on the entire domain and that the convergence rate is geometric provided that each of the subdomain problems is well posed i e their corresponding energy density functions are quasi convex it is shown that the use of a newton type method for the solution of the resultant nonlinear system leads to two kinds of block linearized systems depending on the treatment of the dirichlet boundary conditions the first kind is a symmetric block diagonal linear system in which each diagonal block is the tangent stiffness of each subdomain i e the off diagonal blocks are all zero and the coupling terms appear only on the right hand side the second kind is a nonsymmetric block system with off diagonal coupling terms several variants of the schwarz alternating method are proposed for the first kind of linear system including one in which the schwarz alternating iterations and the newton iterations are combined into a single scheme this version of the method is particularly attractive as it lends itself to a minimally intrusive implementation into existing finite element codes finally we demonstrate the performance of the proposed variants of the schwarz alternating method on several one dimensional and three dimensional examples

on schwarz alternating methods for the incompressible navier stokes equations the schwarz alternating method can be used to solve linear 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 result from solving a sequence of elliptic boundary value problems in each of the subdomains this paper considers four schwarz alternating methods for the n dimensional steady viscous incompressible navier stokes equations n leq 4 it is shown that the schwarz sequences converge to the true solution provided that the reynolds number is sufficiently small

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

monotone iterative algorithms for a nonlinear singularly perturbed parabolic problem this paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed parabolic reaction diffusion problem firstly the monotone method known as the method of lower and upper solutions is applied to computing a nonlinear difference scheme obtained after discretisation of the continuous problem secondly a monotone domain decomposition algorithm based on a modification of the schwarz alternating method is constructed this monotone algorithm solves only linear discrete systems at each iterative step of the iterative process the rate of convergence of the monotone domain decomposition algorithm is estimated numerical experiments are presented

linearized domain decomposition approaches for nonlinear boundary value problems abstract linearized schwarz domain decomposition approaches for nonlinear boundary value problems of the form u f x u u subject to dirichlet boundary conditions are proposed and analyzed in the presence of subsolutions and supersolutions we construct a globally convergent linear monotone iteration suitable for implementation in a distributed computing environment these iterations provide an alternative to the typical locally convergent approach of discretizing and solving the resulting non linear algebraic equations using a newton iteration the work also extends previous results obtained in the case where f has no dependence on the derivative of the solution the schwarz iteration is first proposed and studied in detail on two subdomains the result is then generalized to an arbitrary number of subdomains both alternating and parallel schwarz iterations are analyzed numerical results are provided to demonstrate the theory and the utility of the proposed iterations

block monotone iterative method for semilinear parabolic equations with nonlinear boundary conditions two block monotone iterative schemes called jacobi and gauss seidel monotone iterations are presented for numerical solutions of a class of semilinear parabolic equations under nonlinear boundary conditions by the finite difference method these iteration schemes extend the method for semilinear elliptic boundary value problems to parabolic equations including a comparison result between them it is shown that by using an upper solution and a lower solution as initial iterations each of the iterative schemes yields two sequences which converge monotonically from above and below respectively to a unique solution of the finite difference system some error estimates and a convergence theorem are given and various sufficient conditions for the construction of upper and lower solutions are obtained numerical results are presented for some physical model problems including some problems with known continuous solutions and two problems with l shaped and trapezoidal domains

on schwarz alternating methods for nonlinear elliptic 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 this paper considers several schwarz alternating methods for nonlinear elliptic problems we show that schwarz alternating methods can be embedded in the framework of common techniques such as banach and schauder fixed point methods and global inversion methods used to solve these nonlinear problems

on linear monotone iteration and schwarz methods for nonlinear elliptic 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 subdomain in this paper proofs of convergence of some schwarz alternating methods for nonlinear elliptic 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 some solution on finitely many subdomains even when multiple solutions are possible 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

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

abstract capacitary estimates and the completeness and separability of certain classes of non locally convex topological vector spaces abstract we are concerned with establishing completeness and separability criteria for large classes of topological vector spaces which are typically non locally convex including lebesgue like spaces lorentz spaces orlicz spaces mixed normed spaces tent spaces and discrete triebel lizorkin and besov spaces for vector spaces of measurable functions we also derive pointwise convergence results our approach relies on abstract capacitary estimates and works in certain cases of interest even in the absence of a background measure space and or of a vector space structure

global and uniform convergence of subspace correction methods for some convex optimization problems this paper gives some global and uniform convergence estimates for a class of subspace correction based on space decomposition iterative methods applied to some unconstrained convex optimization problems some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems

on the schwarz alternating method with more than two subdomains for nonlinear monotone problems in a hilbert space an iterative process is defined which generalizes the schwarz alternating method for nonlinear monotone problems to more than two subdomains and the convergence of this process is proved in the case of the problems with the solution in a convex of h 0 1 omega it is proved that the method converges geometrically numerical results are given concerning the problem of elastic plastic torsion of prismatic bars and the problem of seepage of fluids through porous media

rate of convergence of some space decomposition methods for linear and nonlinear problems convergence of a space decomposition method is proved for a class of convex programming problems a space decomposition refers to a method that decomposes a space into a sum of subspaces which could be a domain decomposition or a multilevel method when applied to partial differential equations two algorithms are proposed both can be used for linear as well as nonlinear elliptic problems and they reduce to the standard additive and multiplicative schwarz methods for linear elliptic problems

a block monotone domain decomposition algorithm for a semilinear convection diffusion problem this paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed convection diffusion problem a block monotone domain decomposition algorithm based on a schwarz alternating method and on block iterative scheme is constructed this monotone algorithm solves only linear discrete systems at each iterative step of the iterative process and converges monotonically to the exact solution of the nonlinear problem the rate of convergence of the block monotone domain decomposition algorithm is estimated numerical experiments are presented

boundary element monotone iteration scheme for semilinear elliptic partial differential equations part ii quasimonotone iteration for coupled 2 2 systems numerical solutions of 2 2 semilinear systems of elliptic boundary value problems whose nonlinearities are of quasimonotone nondecreasing quasimonotone nonincreasing or mixed quasimonotone types are computed at each step of the quasi monotone iteration the solution is represented by a simple layer potential plus a domain integral the simple layer density is then discretized by boundary elements because of the various combinations of dirichlet neumann and robin boundary conditions there is an associated 2 2 matrix problem the norm of which must be estimated from the analysis of such 2 2 matrices we formulate conditions which guarantee the monotone iteration a strict contraction staying within the close range of a given pair of subsolution and supersolution thereafter boundary element error analysis can be carried out in a similar way as for the discretized problem a concrete example of a monotone dissipative system on a 2d annular domain is also computed and illustrated

boundary element monotone iteration scheme for semilinear elliptic partial differential equations the monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems with the availability of a supersolution and a subsolution the iterates converge monotonically to one or two solutions of the nonlinear pde however the rates of such monotone convergence cannot be determined in general in addition when the monotone iteration scheme is implemented numerically through the boundary element method error estimates have not been analyzed in earlier studies in this paper we formulate a working assumption to obtain an exponentially fast rate of convergence this allows a margin 6 for the numerical implementation of boundary elements within the range of monotone convergence we then interrelate several approximate solutions and use the aubin nitsche lemma and the triangle inequalities to derive error estimates for the galerkin boundary element iterates with respect to the h r 0 r 2 sobolev space norms such estimates are of optimal order furthermore as a peculiarity we show that for the nonlinearities that are of separable type higher than optimal order error estimates can be obtained with respect to the mesh parameter h several examples of semilinear elliptic partial differential equations featuring different situations of existence nonexistence uniqueness multiplicity and stability are discussed computed and the graphics of their numerical solutions are illustrated