experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

improving newton raphson method for nonlinear equations by modified adomian decomposition method in this paper we present some efficient numerical algorithms for solving nonlinear equations based on newton raphson method the modified adomian decomposition method is applied to construct the numerical algorithms some numerical illustrations are given to show the efficiency of algorithms

an extension of the conjugate residual method to nonsymmetric linear systems the conjugate gradient cg method and the conjugate residual cr method are krylov subspace methods for solving symmetric positive definite linear systems to solve nonsymmetric linear systems the bi conjugate gradient bi cg method has been proposed as an extension of cg bi cg has attractive short term recurrences and it is the basis for the successful variants such as bi cgstab in this paper we extend cr to nonsymmetric linear systems with the aim of finding an alternative basic solver numerical experiments show that the resulting algorithm with short term recurrences often gives smoother convergence behavior than bi cg hence it may take the place of bi cg for the successful variants

the natural element method in solid mechanics the application of the natural element method nem 1 2 to boundary value problems in two dimensional small displacement elastostatics is presented the discrete model of the domain consists of a set of distinct nodes n and a polygonal description of the boundary in the natural element method the trial and test functions are constructed using natural neighbour interpolants these interpolants are based on the voronoi tessellation of the set of nodes n the interpolants are smooth c 1 everywhere except at the nodes where they are c 0 in one dimension nem is identical to linear nite elements the nem interpolant is strictly linear between adjacent nodes on the boundary of the convex hull which facilitates imposition of essential boundary conditions a methodology to model material discontinuities and non convex bodies cracks using nem is also described a standard displacement based galerkin procedure is used to obtain the discrete system of linear equations application of nem to various problems in solid mechanics which include the patch test gradient problems bimaterial interface and a static crack problem are presented excellent agreement with exact analytical solutions is obtained which exemplies the accuracy and robustness of nem and suggests its potential application in the context of other classes of problems crack growth plates and large deformations to name a few 1998 john wiley sons ltd

on new exponential quadratically convergent iterative formulae a class of exponential iterative formulae having quadratic convergence is discussed in this paper the theoretical analysis and numerical experiments show that new exponential iterative formulae are effective and comparable to well known method of newton

meshfree and finite element nodal integration methods nodal integration can be applied to the galerkin weak form to yield a particle type method where stress and material history are located exclusively at the nodes and can be employed when using meshless or finite element shape functions this particle feature of nodal integration is desirable for large deformation settings because it avoids the remapping or advection of the state variables required in other methods to a lesser degree nodal integration can be desirable because it relies on fewer stress point evaluations than most other methods in this work aspects regarding stability consistency efficiency and explicit time integration are explored within the context of nodal integration both small and large deformation numerical examples are provided copyright 2007 john wiley sons ltd

a note on the halley method in banach spaces abstract we provide a convergence theorem under the usual ostrowski kantorovich conditions for an equivalent form of the halley method in banach spaces the best possible upper error bound for the method is obtained some applications to the solution of quadratic operator equations are provided

pressure stability in fractional step finite element methods for incompressible flows abstract the objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows that use a pressure poisson equation for the classical first order projection method it is shown that there is a pressure control which depends on the time step size and therefore there is a lower bound for this time step for stability reasons the situation is much worse for a second order scheme in which part of the pressure gradient is kept in the momentum equation the pressure stability in this case is extremely weak to overcome these shortcomings a stabilized fractional step finite element method is also considered and its stability is analyzed some simple numerical examples are presented to support the theoretical results

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

gpbi cg generalized product type methods based on bi cg for solving nonsymmetric linear systems recently bi cgstab as a variant of bi cg has been proposed for solving nonsymmetric linear systems and its attractive convergence behavior has been confirmed in many numerical experiments bi cgstab can be characterized by its residual polynomial which consists of the product of the residual polynomial of bi cg with other polynomials generated from two term recurrence relations in this paper we propose a unified way to generalize a class of product type methods whose residual polynomials can be factored by the residual polynomial of bi cg and other polynomials with standard three term recurrence relations such product type methods which are based on bi cg can be regarded as generalizations of bi cgstab from the unified way the well known variants of the product type methods like cgs bi cgstab bi cgstab2 are reacquired again

the bicor and cors iterative algorithms for solving nonsymmetric linear systems we present two iterative algorithms for solving real nonsymmetric and complex non hermitian linear systems of equations and that were developed from variants of the nonsymmetric lanczos method in this paper we give the theoretical background of the two iterative methods and discuss their main computational aspects using a large number of numerical experiments we analyze their convergence properties and we also compare them with other popular nonsymmetric iterative solvers in use today

a fe meshfree quad4 element based on partition of unity a new 4 node quadrilateral element is developed by combining the shape functions of finite element and meshfree methods the objective is to synergise the attractive features of finite element and meshfree shape functions using the concepts of partition of unity pu methods the shape functions of the well known quad4 element are used for pu and the least square point interpolation for local approximation this enables the proposed element inherit the compatibility properties of finite element method and the quadratic polynomial completeness properties of meshfree methods numerical results for the bench mark problems show that the proposed element has a good tolerance to distorted meshes the element also seems to be free from the linear dependence problem which is known to plague some of the pu based elements

iterative methods improving newton s method by the decomposition method in this paper we present a sequence of iterative methods improving newton s method for solving nonlinear equations the adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses the orders of convergence are derived analytically and then rederived by applying symbolic computation of maple some numerical illustrations are given

a study on the performance of natural neighbour based galerkin methods in this paper we address the problem of improving natural element simulations in terms of computational cost several problems are discussed that include an efficient natural neighbour search algorithm and a comparison of different natural neighbour based interpolation algorithms in particular we review the so called pseudo nem a moving least squares like approximation scheme that employs natural neighbours and compare it with traditional sibson and laplace interpolation schemes in terms of both accuracy and computational cost some examples in linear elasticity and visco plasticity are analysed in order to test the proposed schemes in engineering problems

a comparative study of iterative solutions to linear systems arising in quantum mechanics this study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by sherry li of nersc numerical experiments show the feasibility of iterative methods to some extent when applied to the problems and reveal the competitiveness of our recently proposed lanczos biconjugate a orthonormalization methods to other classic and popular iterative methods by the way experiment results also indicate that application specific preconditioners may be mandatory and required for accelerating convergence

a partition of unity based fe meshfree quad4 element with radial polynomial basis functions for static analyses abstract recently a partition of unity pu based finite element called fe meshfree quad4 element has been published that element employed polynomial basis functions for the local approximation la in the present paper a new fe meshfree quad4 element employing hybrid radial polynomial basis functions for the la is proposed an advantage of radial polynomial basis is the freedom from the possible singularity of the moment matrix that could sometimes result with an inappropriate choice of polynomial basis functions another advantage is the improved accuracy of finite element solution the new element has been applied to several linear and nonlinear test problems the results demonstrate that the new element gives much better performance as compared to the previous fe meshfree quad4 element with pure polynomial basis even with a lower order basis viz with just three or four polynomial terms the present element is capable of giving more accurate solution than the previous element which employs a six term polynomial basis the new element also exhibits a much higher degree of tolerance to mesh distortion than the known quadratic elements moreover the linear dependence problem otherwise associated with many of the partition of unity pu based elements is completely eliminated from the present element

on newton type methods with cubic convergence recently there has been some progress on newton type methods with cubic convergence that do not require the computation of second derivatives weerakoon and fernando appl math lett 13 2000 87 derived the newton method and a cubically convergent variant by rectangular and trapezoidal approximations to newton s theorem while frontini and sormani j comput appl math 156 2003 345 140 2003 419 derived further cubically convergent variants by using different approximations to newton s theorem homeier j comput appl math 157 2003 227 169 2004 161 independently derived one of the latter variants and extended it to the multivariate case here we show that one can modify the werrakoon fernando approach by using newton s theorem for the inverse function and derive a new class of cubically convergent newton type methods

the meshless finite element method a meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n and the condition that the shape functions depend only on the node positions furthermore the method proposed also shares several of the advantages of the finite element method such as a the simplicity of the shape functions in a large part of the domain b c0 continuity between elements which allows the treatment of material discontinuities and c ease of introduction of the boundary conditions copyright 2003 john wiley sons ltd

a hybridized iterative algorithm of the bicorstab and gpbicor methods for solving non hermitian linear systems in this study we derive a new iterative algorithm including its preconditioned version which is a hybridized variant of the biconjugate a orthogonal residual stabilized bicorstab method and the generalized product type solvers based on bicor gpbicor method the proposed method which is named gpbicor m similarly to the gpbicg m method proposed by fujino 2002 can be regarded as an extension of the bicorstab2 method introduced by zhao and huang 2013 inspired by fujino s idea for improving the bicgstab2 method in the established gpbicor m method the parameters computed by the bicorstab method are chosen at successive m iteration steps and afterwards the parameters of the gpbicor method are utilized in the subsequent iteration steps therefore the proposed method can inherit the low computational cost of bicorstab and the attractive convergence of gpbicor extensive numerical convergence results on selected real and complex matrices are shown to assess the performances of the proposed gpbicor m method also against other popular non hermitian krylov subspace methods

ml k bicgstab a bicgstab variant based on multiple lanczos starting vectors we present a variant of the popular bicgstab method for solving nonsymmetric linear systems the method which we denote by ml k bicgstab is derived from a variant of the bicg method based on a lanczos process using multiple k 1 starting left lanczos vectors compared with the original bicgstab method our new method produces a residual polynomial which is of lower degree after the same number of steps but which also requires fewer matrix vector products to generate on average requiring only 1 1 k matvecs per step empirically it also seems to be more stable and more quickly convergent the new method can be implemented as a k term recurrence and can be viewed as a bridge connecting the arnoldi based fom gmres methods and the lanczos based bicgstab methods

evaluation of high order versions of the diffuse approximate meshless method a high order version of a diffuse approximate meshless method has been implemented and tested on two problems namely diffusion and transport diffusion two dimensional equations it is shown that the fourth order approximation greatly improves the numerical solution

an improved bi conjugate residual algorithm suitable for distributed parallel computing an improved bi conjugate residual bicr method ibicr method in brief for solving large sparse linear systems with unsymmetrical coefficient matrices was proposed for distributed parallel environments the method reduced two global synchronization points to one by reconstructing bicr method and all inner products per iteration were independent and communication time required for inner product can be overlapped efficiently with computation time of vector updates it combines the elements of numerical stability with the characters of design of parallel algorithms the cost is only a little increased computation performance and isoefficiency analysis shows that ibicr method has better parallelism and scalability than bicr method numerical experiments show that the parallel performance can be improved by a factor of about 2 we compared also ibicr with ibicg and bicr with bicg methods the results show that bicr and ibicr methods convergent at the same number of iteration and they convergent faster than bicg and ibicg methods respectively furthermore ibicr method conquers the vibration of residual norm of ibicg method

improving the order of convergence of iteration functions abstract improving methods for the order of convergence of iteration functions are give using these methods new third or fourth order root finding methods for a single equation with a multiple root are derived numerical examples are given

variants of bicgstab for matrices with complex spectrum recently van der vorst siam j sci statist comput 13 1992 pp 631 644 proposed for solving nonsymmetric linear systems az b a biconjugate gradient bicg based krylov space method called bicgstab that like the biconjugate gradient squared bicgs method of sonneveld does not require matrix vector multiplications with the transposed matrix a t and that has typically a much smoother convergence behavior than bicg and bicgs its nth residual polynomial is the product of the one of bicg i e the nth lanczos polynomial with a polynomial of the same degree with real zeros therefore nonreal eigenvalues of a are not approximated well by the second polynomial factor here the author presents for real nonsymmetric matrices a method bicgstab2 in which the second factor may have complex conjugate zeros moreover versions suitable for complex matrices are given for both methods

bicgcr2 a new extension of conjugate residual method for solving non hermitian linear systems abstract in the present paper we introduce a new extension of the conjugate residual cr method for solving non hermitian linear systems with the aim of developing an alternative basic solver to the established biconjugate gradient bicg biconjugate residual bicr and biconjugate a orthogonal residual bicor methods the proposed krylov subspace method referred to as the bicgcr2 method is based on short term vector recurrences and is mathematically equivalent to both bicr and bicor we demonstrate by extensive numerical experiments that the proposed iterative solver has often better convergence performance than bicg bicr and bicor hence it may be exploited for the development of new variants of non optimal krylov subspace methods

a cocr method for solving complex symmetric linear systems the conjugate orthogonal conjugate gradient cocg method has been recognized as an attractive lanczos type krylov subspace method for solving complex symmetric linear systems however it sometimes shows irregular convergence behavior in practical applications in the present paper we propose a conjugate a orthogonal conjugate residual cocr method which can be regarded as an extension of the conjugate residual cr method numerical examples show that cocr often gives smoother convergence behavior than cocg

some third order families of iterative methods for solving nonlinear equations abstract in this paper we present a convenient third order iteration formula improving newton s method for solving nonlinear equations with any two iteration functions of order two the formula provides one parameter third order family of methods each family member is free from second derivatives and per iteration it requires three evaluations of the given function and its derivative several numerical examples are given to illustrate the efficiency and performance of some of the presented methods

reproducing kernel element method part i theoretical formulation in this paper and its sequels we introduce and analyze a new class of methods collectively called the reproducing kernel element method rkem the central idea in the development of the new method is to combine the strengths of both finite element methods fem and meshfree methods two distinguished features of rkem are the arbitrarily high order smoothness and the interpolation property of the shape functions these properties are desirable especially in solving galerkin weak forms of higher order partial differential equations and in treating dirichlet boundary conditions so unlike the fem there is no need for special treatment with the rkem in solving high order equations compared to meshfree methods dirichlet boundary conditions do not present any difficulty in using the rkem a rigorous error analysis and convergence study of the method are presented the performance of the method is illustrated and assessed through some numerical examples

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

some variant of newton s method with third order convergence we present a new modification of the newton s method which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given and the best efficiency in term of function evaluations of two of this new methods is provided

construction of newton like iteration methods for solving nonlinear equations in this paper we present a simple and yet powerful and easily applicable scheme in constructing the newton like iteration formulae for the computation of the solutions of nonlinear equations the new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations it provides a tool to develop new newton like iteration methods or to improve the existing iteration methods which contains the well known newton iteration formula in logic those all improve the newton method the orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of maple some numerical tests are given to support the theory developed in this paper

reproducing kernel element method part iii generalized enrichment and applications in this part of the work a notion of generalized enrichment is proposed to construct the global partition polynomials or to enrich global partition polynomial basis with extra terms corresponding to the higher order derivatives of primary variable this is accomplished by either multiplying enrichment functions with the original global partition polynomials or increasing the order of global partition polynomials in the same mesh without refining mesh high order consistency in interpolation hierarchy with generalized kronecker delta property can be straightforwardly achieved in quadrilateral and triangular mesh in 2d by the proposed scheme comparing with the traditional finite element methods the construction proposed here has more flexibility and only needs minimal degrees of freedom the optimal element with high reproducing capacity and overall minimal degrees of freedom can be constructed by the generalized enrichment procedure two optimal elements in two dimensional space have been constructed t10p3i 4 triangular element satisfies third order consistency condition with only 10 degrees of freedom and q15p4i 4 quadrilateral element satisfies fourth order consistency condition with 15 degrees of freedom the performance of interpolation hierarchy is

idr s a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations we present idr s a new family of efficient short recurrence methods for large nonsymmetric systems of linear equations the new methods are based on the induced dimension reduction idr method proposed by sonneveld in 1980 idr s generates residuals that are forced to be in a sequence of nested subspaces although idr s behaves like an iterative method in exact arithmetic it computes the true solution using at most n n s matrix vector products with n the problem size and s the codimension of a fixed subspace we describe the algorithm and the underlying theory and present numerical experiments to illustrate the theoretical properties of the method and its performance for systems arising from different applications our experiments show that idr s is competitive with or superior to most bi cg based methods and outperforms bi cgstab when s 1

a new family of global methods for linear systems with multiple right hand sides the global bi conjugate gradient gl bcg method is an attractive matrix krylov subspace method for solving nonsymmetric linear systems with multiple right hand sides but it often show irregular convergence behavior in many applications in this paper we present a new family of global a biorthogonal methods by using short two term recurrences and formal orthogonal polynomials which contain the global bi conjugate residual gl bcr algorithm and its improved version finally numerical experiments illustrate that the proposed methods are highly competitive and often superior to originals

a finite point method in computational mechanics applications to convective transport and fluid flow the paper presents a fully meshless procedure fo solving partial differential equations the approach termed generically the finite point method is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals some examples showing the accuracy of the method for solution of adjoint and non self adjoint equations typical of convective diffusive transport and also to the analysis of compressible fluid mechanics problem are presented

lagrangian formulations to solve free surface incompressible inviscid fluid flows abstract a particle method is presented for the solution of the incompressible inviscid fluid flow equation using a lagrangian formulation the interpolated function are those used in meshless approximations and the time integration is introduced in a semi implicit way by a fractional step method in this manner both classical stabilization terms used in incompressible euler equations are unnecessary numerical diffusion for convective terms are unnecessary due to the lagrangian formulation and stabilization of pressure due to the incompressibility constraint for equal order interpolations is eliminated using the fractional step method

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

a quasi minimal residual variant of the bi cgstab algorithm for nonsymmetric systems motivated by a recent method of freund siam j sci comput 14 1993 pp 470 482 who introduced a quasi minimal residual qmr version of the conjugate gradients squared cgs algorithm a qmr variant of the biconjugate gradient stabilized bi cgstab algorithm of van der vorst that is called qmrcgstab is proposed for solving nonsymmetric linear systems the motivation for both qmr variants is to obtain smoother convergence behavior of the underlying method the authors illustrate this by numerical experiments that also show that for problems on which bi cgstab performs better than cgs the same advantage carries over to qmrcgstab

a new continuation newton like method and its deformation this paper presents a family of new continuation newton like methods and its deformations for computing approximate solutions of the non linear algebraic equation f x 0 from a practical point of view the new methods are vast superior

a partition of unity based fe meshfree quad4 element for geometric non linear analysis the recently published fe meshfree quad4 element is extended to geometrical non linear analysis the shape functions for this element are obtained by combining meshfree and finite element shape functions the concept of partition of unity pu is employed for the purpose the new shape functions inherit their higher order completeness properties from the meshfree shape functions and the mesh distortion tolerant compatibility properties from the finite element fe shape functions updated lagrangian formulation is adopted for the non linear solution several numerical example problems are solved and the performance of the element is compared with that of the well known q4 qm6 and q8 elements the results show that for regular meshes the performance of the element is comparable to that of qm6 and q8 elements and superior to that of q4 element for distorted meshes the present element has better mesh distortion tolerance than q4 qm6 and q8 elements copyright 2009 john wiley sons ltd

a simply constructed third order modifications of newton s method in this paper we present a simple and easily applicable approach to construct some third order modifications of newton s method for solving nonlinear equations it is shown by way of illustration that existing third order methods can be employed to construct new third order iterative methods the proposed approach is applied to the classical chebyshev halley methods to derive their second derivative free variants numerical examples are given to support that the methods thus obtained can compete with known third order methods

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

the particle finite element method a powerful tool to solve incompressible flows with free surfaces and breaking waves summary particle methods are those in which the problem is represented by a discrete number of particles each particle moves accordingly with its own mass and the external internal forces applied to it particle methods may be used for both discrete and continuous problems in this paper a particle method is used to solve the continuous fluid mechanics equations to evaluate the external applied forces on each particle the incompressible navier stokes equations using a lagrangian formulation are solved at each time step the interpolation functions are those used in the meshless finite element method and the time integration is introduced by an implicit fractional step method in this manner classical stabilization terms used in the momentum equations are unnecessary due to lack of convective terms in the lagrangian formulation once the forces are evaluated the particles move independently of the mesh all the information is transmitted by the particles fluid structure interaction problems including free fluid surfaces breaking waves and fluid particle separation may be easily solved with this methodology copyright 2004 john wiley sons ltd

bicr variants of the hybrid bicg methods for solving linear systems with nonsymmetric matrices we propose bi conjugate residual bicr variants of the hybrid bi conjugate gradient bicg methods referred to as the hybrid bicr variants for solving linear systems with nonsymmetric coefficient matrices the recurrence formulas used to update an approximation and a residual vector are the same as those used in the corresponding hybrid bicg method but the recurrence coefficients are different they are determined so as to compute the coefficients of the residual polynomial of bicr from our experience it appears that the hybrid bicr variants often converge faster than their bicg counterpart numerical experiments show that our proposed hybrid bicr variants are more effective and less affected by rounding errors the factor in the loss of convergence speed is analyzed to clarify the difference of the convergence between our proposed hybrid bicr variants and the hybrid bicg methods

lanczos type variants of the cocr method for complex nonsymmetric linear systems motivated by the celebrated extending applications of the well established complex biconjugate gradient cbicg method to deal with large three dimensional electromagnetic scattering problems by pocock and walker m d pocock s p walker the complex bi conjugate gradient solver applied to large electromagnetic scattering problems computational costs and cost scalings ieee trans antennas propagat 45 1997 140 146 three lanczos type variants of the recent conjugate a orthogonal conjugate residual cocr method of sogabe and zhang t sogabe s l zhang a cocr method for solving complex symmetric linear systems j comput appl math 199 2007 297 303 are explored for the solution of complex nonsymmetric linear systems the first two can be respectively considered as mathematically equivalent but numerically improved popularizing versions of the bicr and crs methods for complex systems presented in sogabe s ph d dissertation and the last one is somewhat new and is a stabilized and more smoothly converging variant of the first two in some circumstances the presented algorithms are with the hope of obtaining smoother and hopefully faster convergence behavior in comparison with the cbicg method as well as its two corresponding variants this motivation is demonstrated by numerical experiments performed on some selective matrices borrowed from the university of florida sparse matrix collection by davis

on a class of quadratically convergent iteration formulae in this paper a new class of iterative formulae having quadratic convergence is presented furthermore these algorithms are comparable to the well known method of newton and the computed results support this theory

bicr type methods for families of shifted linear systems the shifted linear systems with non hermitian matrices often arise from the numerical solutions for time dependent pdes computing the large scale eigenvalue problems control theory and so on in the present paper we develop two shifted variants of bicr type methods for solving such linear systems these variants of bicr type methods take advantage of the shifted structure so that the number of matrix vector multiplications and the number of inner products are the same as a single linear system finally extensive numerical examples are reported to illustrate the performance and effectiveness of the proposed methods

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

a generalized product type bicor method and its application in signal deconvolution for solving nonsymmetric linear systems we attempt to establish symmetric structures in nonsymmetric systems and handle them through the methods devised for symmetric cases a biconjugate a orthogonal residual method based on biconjugate a orthonormalization procedure has been proposed and nominated as bicor in y f jing t z huang y zhang l li g h cheng z g ren y duan t sogabe b carpentieri lanczos type variants of the cocr method for complex nonsymmetric linear systems j comput phys 228 2009 6376 6394 as many similar characteristics exist between bicor and bicg the strategies of improved variants of bicg such as cgs and bicgstab can be utilized to enhance the algorithm for bicor making use of the product of residual polynomials of bicor and other polynomials cors and bicorstab have been proposed along the same ideas of cgs and bicgstab respectively in the above mentioned paper in this paper a unified generalized framework of product type bicor which is epitomized by the product of residual polynomials and other polynomials is proposed numerical examples are selected from the blurring signal cases and the effect of the generalized product type bicor method is prominent in signal deconvolution

a stabilized finite point method for analysis of fluid mechanics problems in this paper a meshless procedure termed the finite point method for solving convection diffusion and fluid flow type problems is presented the method is based on the use of a weighted least square interpolation procedure together with point collocation for evaluating the approximation integrals special emphasis is given to the stabilization of the convective terms and the neumann boundary condition which has been found to be essential to obtain accurate results some examples of application to diffusive and convective transport and compressible flow problems using quadratic fp interpolations are presented

to mesh or not to mesh that is the question in the last decade a family of methods called meshless methods has been developed both for structural and fluid mechanics problems after these ideas a possible classification for numerical formulations may be to separate the methods that make use of a standard finite element mesh such as those made of tetrahedra or hexahedra from those that do not need a standard mesh namely the meshless methods for solving a partial different equation by a numerical method a possible alternative may be either to use a mesh method or a meshless method this paper discusses this issue to show that this choice is not in the large majorities of the cases the right question

adaptive meshless centres and rbf stencils for poisson equation we consider adaptive meshless discretisation of the dirichlet problem for poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions new meshless stencil selection and adaptive refinement algorithms are proposed in 2d numerical experiments show that the accuracy of the solution is comparable with and often better than that achieved by the mesh based adaptive finite element method

a stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation abstract a stabilized finite element formulation for incompressible viscous flows is derived the starting point are the modified navier stokes equations incorporating naturally the necessary stabilization terms via a finite increment calculus fic procedure application of the standard finite element galerkin method to the modified differential equations leads to a stabilized discrete system of equations overcoming the numerical instabilities emanating from the advective terms and those due to the lack of compatibility between approximate velocity and pressure fields the fic method also provides a natural explanation for the stabilization terms appearing in all equations for both the navier stokes and the simpler stokes equations transient solution schemes with enhanced stabilization properties are also proposed finally a procedure for computing the stabilization parameters is presented

polyhedrization of an arbitrary 3d point set abstract given a 3d point set the problem of defining the volume associated dividing it into a set of regions elements and defining a boundary surface is tackled several physical problems need to define volume domains boundary surfaces and approximating functions from a given point distribution this is for instance the case of particle methods in which all the information is the particle positions and there are not boundary surfaces definition until recently all the fem mesh generators were limited to the generation of simple elements as tetrahedral or hexahedral elements or triangular and quadrangular in 2d problems the reason of this limitation was the lack of any acceptable shape function to be used in other kind of geometrical elements nowadays there are several acceptable shape functions for a very large class of polyhedra these new shape functions together with a generalization of the delaunay tessellation presented in this paper gives an optimal marriage and a powerful tool to solve a large variety of physical problems by numerical methods the domain partition into polyhedra presented here is not a standard mesh generation the problem here is for a given node distribution to find a suitable boundary surface and a suitable mesh to be used in the solution of a physical problem by a numerical method to include new nodes or change their positions is not allowed