a finite element method using singular functions for the poisson equation corner singularities consider the poisson equation with homogeneous dirichlet boundary conditions on a polygonal domain with one reentrant corner in this paper we develop a new finite element method for the accurate computation of the solution and stress intensity factors it is well known that the solution of such a problem has a singular function representation u w lambda eta s where w in h 2 omega cap h 1 0 omega lambda in cal r and eta are the stress intensity factor and cut off function respectively and s is a known singular function depending only on the reentrant angle by using the dual singular and an extra cut off function we are able to derive a new extraction formula for lambda in terms of w and hence deduce a well posed variational problem for w standard continuous piecewise linear finite element approximation yields o h optimal accuracy for w which in turn implies the same accuracy for u in the h 1 norm we are able only to prove o h 1 frac pi omega error bounds for w and u in the l 2 norm and for lambda in the absolute value where omega is the internal angle

a finite element method for computing accurate solutions for poisson equations with corner singularities using the stress intensity factor in this article we consider the poisson equation with homogeneous dirichlet boundary conditions on a polygonal domain with one reentrant corner the solution of the poisson equation with a concave corner yields a singular decomposition u w s where w is regular s is a singular function and the coefficient is the so called stress intensity factor this stress intensity factor can be computed using the extraction formula we introduce a new non homogeneous boundary value problem which has zero stress intensity factor using the solution of this new partial differential equation we can compute an accurate solution of the original problem simply by adding singular part we obtain an optimal convergence rate with smaller errors when compared with others

group accelerated overrelaxation methods on rotated grid in martins et al m m martins w s yousif d j evans explicit group aor method for solving elliptic partial differential equations neural parallel and science computation 10 4 2002 411 422 a new explicit four point group accelerated overrelaxation group aor iterative method was presented where the computational superiority of this new technique was established when compared with the point aor method developed by evans and martins d j evans m m martins the aor method for ax xb c international journal of computer mathematics 52 1994 75 82 and martins et al m m martins d j evans m e trigo the aor iterative method for new preconditioned linear systems journal of computational and applied mathematics 132 2001 461 466 in this work we formulate an alternative group scheme from the aor family derived from the rotated skewed five point formula g dahlquist a bjorck numerical methods prentice hall englewood cliffs nj 1974 p 320 y saad iterative methods for sparse linear systems second ed pws publishing company boston 2000 p 50 the derivation of this new group scheme is presented and its performance is compared with the existing explicit four point group aor we include the analysis of the convergence results for the new group scheme numerical experiments are also presented to illustrate our results

optimal a priori estimates for interface problems we consider a priori estimates in weighted norms for interface problems with piecewise constant diffusion constants which do not depend on the ratio between the constants our result generalizes an estimate of lemrabet to arbitrary dimensions and includes curved boundaries furthermore we discuss criteria for the existence of a uniform poincare estimate in weighted norms in the affirmative case we obtain a robust finite element error bound in weighted norms finally we present numerical experiments including a case with no uniform poincare constant

nested newton strategies for energy corrected finite element methods energy corrected finite element methods provide an attractive technique for dealing with elliptic problems in domains with re entrant corners optimal convergence rates in weighted l 2 norms can be fully recovered by a local modification of the stiffness matrix at the re entrant corner and no pollution effect occurs although the existence of optimal correction factors is established it remains open how to determine these factors in practice first we show that asymptotically a unique correction parameter exists and that it can be formally obtained as the limit of level dependent correction parameters which are defined as roots of an energy defect function second we propose three nested newton type algorithms using only one newton step per refinement level and show local or even global convergence to this asymptotic correction parameter

energy corrected finite element methods for corner singularities it is well known that the regularity of solutions of elliptic partial differential equations on domains with re entrant corners is limited by the maximal interior angle this results in reduced convergence rates for finite element approximations on families of quasi uniform meshes following an idea of zenger and gietl we show that it is possible to regain the full order of convergence by a local modification of the bilinear form in a vicinity of the singularity and thus to overcome the pollution effect a complete convergence analysis in weighted sobolev spaces is presented and we also show that the stress intensity factors can be computed with optimal accuracy the theoretical results are illustrated by numerical tests that demonstrate second order convergence of linear finite elements on families of quasi uniform meshes

the fourier singular complement method for the poisson problem part ii axisymmetric domains this paper is the second part of a threefold article aimed at solving numerically the poisson problem in three dimensional prismatic or axisymmetric domains in the first part of this series the fourier singular complement method was introduced and analysed in prismatic domains in this second part the fscm is studied in axisymmetric domains with conical vertices whereas in the third part implementation issues numerical tests and comparisons with other methods are carried out the method is based on a fourier expansion in the direction parallel to the reentrant edges of the domain and on an improved variant of the singular complement method in the 2d section perpendicular to those edges neither refinements near the reentrant edges or vertices of the domain nor cut off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of fourier modes used

an error analysis of the finite element method overcoming corner singularities for the stationary stokes problem in this paper we introduce a mixed finite element method to overcome corner singularities of the stationary stokes system on a non convex polygon the reference choi and kweon 2013 says that the velocity field and the pressure function are composed of singular parts and regular remainders near a non convex vertex and furthermore the regular parts are sufficiently smooth on the polygon we use the corner singularity expansion and propose new extraction formulae for coefficients of singularities for the proposed numerical method we first try to find finite element solutions for the regular parts and then calculate approximations of the coefficients by the derived formulae expressed by the remainders and given functions we show error estimates of the approximations for the regular parts and coefficients and give some numerical experiments to confirm the efficiency and reliability of the proposed method

error estimate of a finite element method using stress intensity factor abstract an algorithm on computing accurate finite element approximation to the poisson equation on a polygonal domain with corner singularities was studied in kim and lee 2016 2017 numerically the algorithm requires several iterations depending on singularities of the solution each iteration requires a solution of the standard finite element approximation to the poisson equation with possible different dirichlet data and the corresponding stress intensity factors this paper provides an error estimate of the finite element approximation given by the algorithm and hence determine the number of iterations needed to achieve full rates of convergence in both the energy and the l 2 norms

a finite element method for elliptic optimal control problem on a non convex polygon with corner singularities abstract in this paper we study a finite element method overcoming corner singularities for elliptic optimal control problem posed on a polygon based on a corner singularity decomposition of singular solution for the associated elliptic system we derive an extraction formula for coefficients of singular parts called stress intensity factors and introduce a finite element discretization for regular parts of the solution pair furthermore we analyze the finite element solutions in l 2 and h 1 norms by numerical experiments we confirm the efficiency and reliability of the proposed method

the aor iterative method for new preconditioned linear systems 461 abstract in this paper we apply the aor method to preconditioned linear systems different from those considered in evans and martins internat j comput math 5 1995 69 76 gunawardena et al linear algebra appl 154 156 1991 123 143 and li and evans technical report no 901 department of computer studies university of loughborough 1994 our results show that some improvements in the convergence rate of this iterative method can be obtained

numerical performance of parallel group explicit solvers for the solution of fourth order elliptic equations many applications in applied mathematics and engineering involve numerical solutions of partial differential equations pdes various discretisation procedures such as the finite difference method result in a problem of solving large sparse systems of linear equations in this paper a group iterative numerical scheme based on the rotated skewed five point finite difference discretisation is proposed for the solution of a fourth order elliptic pde which represents physical situations in fluid mechanics and elasticity the rotated approximation formulas lead to schemes with lower computational complexities compared to the centred approximation formulas since the iterative procedure need only involve nodes on half of the total grid points in the solution domain we describe the development of the parallel group iterative scheme on a cluster of distributed memory parallel computer using message passing interface mpi programming environment a comparative study with another group iterative scheme derived from the centred difference formula is also presented a detailed performance analysis of the parallel implementations of both group methods will be reported and discussed

analysis of first order system least squares fosls for elliptic problems with discontinuous coefficients part i first order system least squares fosls is a recently developed methodology for solving partial differential equations among its advantages are that the finite element spaces are not restricted by the inf sup condition imposed for example on mixed methods and that the least squares functional itself serves as an appropriate error measure this paper studies the fosls approach for scalar second order elliptic boundary value problems with discontinuous coefficients irregular boundaries and mixed boundary conditions a least squares functional is defined and ellipticity is established in a natural norm of an appropriately scaled least squares bilinear form for some geometries this ellipticity is independent of the size of the jumps in the coefficients the occurrence of singularities at interface corners cross points reentrant corners and irregular boundary points is discussed and a basis of singular functions with local support around singular points is established a companion paper shows that the singular basis functions can be added at little extra cost and lead to optimal performance of standard finite element discretization and multilevel solver techniques also independent of the size of coefficient jumps for some geometries

asymptotic error rates in quantum hypothesis testing we consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero this leads to the identification of the quantum generalisation of the classical chernoff distance which is the corresponding quantity in classical symmetric hypothesis testing

weighted norm first order system least squares fosls for problems with corner singularities a weighted norm least squares method is considered for the numerical approximation of solutions that have singularities at the boundary while many methods suffer from a global loss of accuracy due to boundary singularities the least squares method can be particularly sensitive to a loss of regularity the method we describe here requires only a rough lower bound on the power of the singularity and can be applied to a wide range of elliptic equations optimal order discretization accuracy is achieved in weighted h 1 and functional norms and l 2 accuracy are retained for boundary value problems with a dominant div curl operator our analysis including interpolation bounds and several poincare type inequalities are carried out in appropriately weighted sobolev spaces numerical results confirm the error bounds predicted in the analysis

multigrid methods for the computation of singular solutions and stress intensity factors i corner singularities we consider the poisson equation u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain with cracks multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed the convergence rate for the stress intensity factors is mathcal o h 3 2 in whenfel2 and mathcal o h 2 in whenfeh1 the convergence rate in the energy norm is mathcal o h 1 in in the first case and mathcal o h in the second case the costs of these multigrid methods are proportional to the number of elements in the triangulation the general case wherefehm is also discussed

an asymptotic fitting finite element method with exponential mesh refinement for accurate computation of corner eddies in viscous flows it is well known that any viscous fluid flow near a corner consists of infinite series of eddies with decreasing size and intensity unless the angle is larger than a certain critical angle h k moffat j fluid mech 18 1964 pp 1 18 the objective of the current work is to simulate such infinite series of eddies occurring in steady flows in domains with corners the problem is approached by high order finite element method with exponential mesh refinement near the corners coupled with analytical asymptotics of the flow near the corners such approach allows one to compute position and intensity of the eddies near the corners in addition to the other main features of the flow the method was tested on the problem of the lid driven cavity flow as well as on the problem of the backward facing step flow the results of computations of the lid driven cavity problem show that the proposed method computes the central eddy with accuracy comparable to the best of existing methods and is more accurate for computing the corner eddies than the existing methods the results also indicate that the relative error of finding the eddies intensity and position decreases uniformly for all the eddies as the mesh is refined i e the relative error in computation of different eddies does not depend on their size

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

a numerical solution for the inhomogeneous dirichlet boundary value problem on a non convex polygon abstract in this paper we introduce an effective finite element scheme for the poisson problem with inhomogeneous dirichlet boundary condition on a non convex polygon due to the corner singularities the solution is composed of a singular part not belonging to the sobolev space h2 and a smoother regular part we first provide a generalized extraction formula for the coefficient of singular part which is depending on the inhomogeneous boundary condition and the regular part we then propose a stable finite element method for the regular part by the use of the derived formula we show the h1 and l2 error estimates of the finite element solution for the regular part and the absolute error estimate of the approximation for the coefficient of singular part finally we give some numerical examples to confirm the efficiency and reliability of the proposed method

stability and convergence of fully discrete galerkin fems for the nonlinear thermistor equations in a nonconvex polygon in this paper we establish the unconditional stability and optimal error estimates of a linearized backward euler galerkin finite element method fem for the time dependent nonlinear thermistor equations in a two dimensional nonconvex polygon due to the nonlinearity of the equations and the non smoothness of the solution in a nonconvex polygon the analysis is not straightforward while most previous efforts for problems in nonconvex polygons mainly focused on linear models our theoretical analysis is based on an error splitting proposed in 30 31 together with rigorous regularity analysis of the nonlinear thermistor equations and the corresponding iterated time discrete elliptic system in a nonconvex polygon with the proved regularity we establish the stability in l infty l infty l l and the convergence in l infty l 2 l l2 for the fully discrete finite element solution without any restriction on the time step size the approach used in this paper may also be applied to other nonlinear parabolic systems in nonconvex polygons numerical results confirm our theoretical analysis and show clearly that no time step condition is needed

energy corrected fem for optimal dirichlet boundary control problems in the presence of re entrant corners the solution of pde constrained optimization problems in general has singular components even when the given data are smooth consequently the use of standard finite element discretization gives rise to the so called pollution effect this means that only a reduced convergence rate compared to the best approximation error is obtained we discuss how optimal convergence rates in weighted norms can be achieved using the idea of energy corrected finite element methods applied to optimal dirichlet boundary control problem in the energy space we present optimal error estimates in weighted norms for the state variable and for the control several numerical examples illustrate the obtained theoretical results

multigrid methods for the computation of singular solutions and stress intensity factors iii interface singularities it is shown in this paper that two dimensional interface problems with large jumps in the coefficients can be solved effectively by the piecewise linear finite element method on quasi uniform grids this is achieved by combining the full multigrid methodology the singular function representation of the solution and the extraction formulas for stress intensity factors the robustness of the algorithm is demonstrated by numerical experiments

a finite element method for singular solutions of the navier stokes equations on a non convex polygon it is shown in choi and kweon 2013 that a solution of the navier stokes equations with no slip boundary condition on a non convex polygon can be written as u p c 1 1 1 c 2 2 2 u r p r near each non convex vertex where u r p r h 2 h 1 i i are corner singularity functions for the stokes problem with no slip condition and c i r are coefficients which are called the stress intensity factors we design a finite element method to approximate the coefficients c i and the regular part u r p r show the unique existence of the approximations and derive their error estimates some numerical examples are given confirming convergence rates for the approximations

first order system ll fosll for general scalar elliptic problems in the plane this paper develops new first order system ll fosll formulations for scalar elliptic partial differential equations it extends the work of z cal et al siam j numer anal 39 2001 pp 1418 1445 where the fosll methodology was first introduced one focus of that paper was to develop fl formulations that allow the use of h1 conforming finite element spaces and optimal multigrid solution techniques to construct l2 approximations of the dependent variables in the presence of discontinuous coefficients the problems for which this goal was achieved were limited to those with no reaction term and with dirichlet and neumann boundaries that were individually connected that is each had at most one component here new fosll formulations are developed to achieve the same goals on a wider class of problems including problems with reaction terms dirichlet and neumann boundaries with multiple components reentrant corners and points at which dirichlet and neumann boundaries meet with an inner angle greater than pi 2 the efficiency of the improved fosll formulations is illustrated by a series of numerical examples

the convergence of multi level methods for solving finite element equations in the presence of singularities the known convergence proofs for multi level methods assume the quasi uniformity of the family of domain triangulations used such triangulations are not suitable for problems with singularities caused by re entrant corners and abrupt changes in the boundary condi tions in this paper it is shown that families of properly refined grids yield the same convergence behavior of multi level methods for such singular problems as quasi uniform subdivisions do for h2 regular problems 1 the continuous problem its discretization and the multi level method using multi level techniques 1 3 4 6 7 8 10 it is possible to solve the large systems of linear equations arising in connection with finite element methods with an amount of work roughly proportional to the number of unknowns this property makes multi level methods at least theoretically superior to all other solution methods including fast solvers based on fft like algorithms which may be directly applied to special problems only the convergence of multi level methods was proved for example by nicolaides 10 bank and dupont 4 and hackbusch 8 all these proofs assume a certain amount of elliptic regularity of the continuous problem to be solved approximately and quasi uniform subdivisions of the domain in finite elements assuming h1 a regularity such quasi uniform triangulations and a jacobi like smoothing procedure bank and dupont 4 and hackbusch 8 showed the following result the rate of convergence of a full iteration step of the multi level method behaves like o m 0 2 uniformly in the number of levels for a growing number m of smoothing steps per level in the optimal case a 1 the problem has to be h2 regular this means for example that the region is not allowed to have re entrant corners if this condition is violated the convergence rate of the multi level procedure decreases and in addition the approximation properties of the finite element discretization itself change for the worse because of the presence of singularities in the solution not captured by the quasi uniform grids the strongly nonuniform systematically re fined triangulations suitable for these problems are not included in the theory so far the aim of the present paper is to fill this gap

a simple finite element method for boundary value problems with a riemann liouville derivative we consider a boundary value problem involving a riemann liouville fractional derivative of order 3 2 2 on the unit interval 0 1 the standard galerkin finite element approximation converges slowly due to the presence of singularity term x 1 in the solution representation in this work we develop a simple technique by transforming it into a second order two point boundary value problem with nonlocal low order terms whose solution can reconstruct directly the solution to the original problem the stability of the variational formulation and the optimal regularity pickup of the solution are analyzed a novel galerkin finite element method with piecewise linear or quadratic finite elements is developed and l 2 d error estimates are provided the approach is then applied to the corresponding fractional sturm liouville problem and error estimates of the eigenvalue approximations are given extensive numerical results fully confirm our theoretical study

optimal order nonnested multigrid methods for solving finite element equations iii on degenerate meshes in this paper we consider several model problems where finite element triangular meshes with arbitrarily small angles high aspect ratios are utilized to deal with anisotropy interfaces or singular perturbations the constant rate independent of the number of unknowns the smallest angle the interface discontinuity the singular perturbation parameter etc convergence of some special nonnested multigrid methods for solving the finite element systems on such degenerate meshes will be proved numerical data are provided to support the analysis in each case

convergence of the multigrid v cycle algorithm for second order boundary value problems without full elliptic regularity the multigrid v cycle algorithm using the richardson relaxation scheme as the smoother is studied in this paper for second order elliptic boundary value problems the contraction number of the v cycle algorithm is shown to improve uniformly with the increase of the number of smoothing steps without assuming full elliptic regularity as a consequence the v cycle convergence result of braess and hackbusch is generalized to problems without full elliptic regularity

the fourier finite element method for the poisson problem on a non convex polyhedral cylinder we study the poisson problem with zero boundary datum in a finite polyhedral cylinder with a non convex edge applying the fourier sine series to the equation along the edge and by a corner singularity expansion for the poisson problem with parameter we define the edge flux coefficient and the regular part of the solution on the polyhedral cylinder we present a numerical method for approximating the edge flux coefficient and the regular part and show the stability we derive an error estimate and give some numerical experiments

the fourier finite element method for the corner singularity expansion of the heat equation near the non convex vertex the solution of the heat equation is of the form u c e r sin w w l 2 r h 2 where c is the stress intensity function of the time variable t the convolution e x t r e r 2 4 t 2 t 3 a cutoff function and the opening angle of the vertex in this paper we use the fourier finite element method for approximating the stress intensity function c and the regular part w and derive the error estimates depending on the regularities of c and w we give some numerical examples confirming the derived convergence rates

a nonconforming generalized finite element method for transmission problems we obtain quasi optimal rates of convergence for transmission interface problems on domains with smooth curved boundaries using a nonconforming generalized finite element method gfem more precisely we study the strongly elliptic problem pu sum partial j a ij partial i u f in a smooth bounded domain omega with dirichlet boundary conditions the coefficients a ij are piecewise smooth possibly with jump discontinuities along a smooth closed surface gamma called the interface which does not intersect the boundary of the domain we consider a sequence of approximation spaces s mu satisfying two conditions 1 nearly zero boundary and interface matching and 2 approximability which are similar to those in babuska nistor and tarfulea j comput appl math 218 2008 pp 175 183 then if u mu in s mu mu ge 1 is a sequence of galerkin approximations of the solution u to the interface problem the approximation error u u mu hat h 1 omega is

the analysis of smoothers for multigrid algorithms the purpose of this paper is to provide a general technique for defining and analyzing smoothing operators for use in multigrid algorithms the smoothing operators considered are based on subspace decomposition and include point line and block versions of jacobi and gauss seidel iteration as well as generalizations we shall show that these smoothers will be effective in multigrid algorithms provided that the subspace decomposition satisfies two simple conditions in many applications these conditions are trivial to verify

solution methods for the poisson equation with corner singularities numerical results in z cai and s kim siam j numer anal 39 2001 pp 286 299 we developed a new finite element method using singular functions for the poisson equation on a polygonal domain with re entrant angles such a method first computes the regular part of the solution then the stress intensity factor and finally the solution itself this paper studies solution methods for solving the system of linear equations arising from the discretization and focuses on numerical results including the finite element accuracy and the multigrid performance

coefficients of the singularities for elliptic boundary value problems on domains with conical points iii finite element methods on polygonal domains in the two first parts of this work rairo model math anal numer 24 1990 pp 27e52 rairo model math anal numer 24 1990 pp 343 367 formulas giving the coefficients arising in the singular expansion of the solutions of elliptic boundary value problems on nonsmooth domains are investigated now for the case of homogeneous strongly elliptic operators with constant coefficients on polygonal domains the solution of such problems by the finite element method is considered in order to approximate the solution or the coefficients different methods are used based on the expressions of the coefficients that were obtained in the first two parts the dual singular function method is also generalized

optimal order nonnested multigrid methods for solving finite element equations i on quasi uniform meshes we prove that the multigrid method works with optimal computational order even when the multiple meshes are not nested when a coarse mesh is not a submesh of the finer one the coarse level correction usually does not have the a projection property and does amplify the iterative error in some components nevertheless the low frequency components of the error can still be caught by the coarse level correction since the amplified highfrequency errors will be damped out by the fine level smoothing efficiently the optimal work order of the standard multigrid method can still be maintained however unlike the case of nested meshes a nonnested multigrid method with one smoothing does not converge in general no matter whether it is a v cycle or a w cycle method it is shown numerically that the convergence rates of nonnested multigrid methods are not necessarily worse than those of nested ones since nonnested multigrid methods accept quite arbitrarily related meshes we may then combine the efficiencies of adaptive refinements and of multigrid algorithms

numerical analysis of singularities in two dimensions part 1 computation of eigenpairs a numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second order elliptic partial differential equations in two dimensions in the vicinity of singular points the singularities may be caused by re entrant corners and abrupt changes in boundary conditions or material properties such singularities are of great interest from the point of view of failure initiation the eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes for this reason failure theories directly or indirectly involve the eigenpairs and their amplitudes this paper addresses the problem of determining the eigenpairs numerically on the basis of the steklov formulation numerical results are presented for several cases importantly the method is applicable to three dimensional cases

predictor corrector p and hp versions of the finite element method for poisson s equation in polygonal domains abstract we consider boundary value problems for the poisson equation on polygonal domains with general nonhomogeneous mixed boundary conditions and derive on the one hand explicit extraction formulas for the coefficients of the singularities on the other hand the formulas are used to construct efficient adaptations for the h p and h p versions of the finite element method for the numerical treatment a priori error estimates show that the h version of the finite element algorithm exhibits the same rate of convergence as it is known for problems with smooth solutions however the principal results of the present work are the robust exponential convergence results for the p and h p versions of the finite element method on quasiuniform meshes in fact it is shown that if the input data source term and boundary data are piecewise analytic then with appropriate choices of conforming finite element subspaces v n of dimension n n the p and h p versions of the finite element algorithms on quasiuniform meshes yield approximate solutions u n to the exact solution u that satisfy the estimates u u n h 1 c 1 e b 1 n 2 3 and u u n h 1 c 2 e b 2 n 1 3 respectively several numerical experiments are included to illustrate the practical effectiveness of the proposed algorithms the results show that the theoretical error analyses are attained within the range of engineering accuracy

sharp estimates for multigrid rates of convergence with general smoothing and acceleration in this paper we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations our theory is presented entirely in terms of the generalized condition number kappa of the matrix a and the smoothing matrix b this leads to a completely algebraic analysis of the method as an iterative technique for solving linear equations the properties of the elliptic equation and discretization procedure enter only when we seek to estimate kappa just as in the case of most standard iterative methods here we consider the fundamental two level iteration and the v and w cycles of the j level iteration j 2 we prove that the v and w cycles converge even when only one smoothing iteration is used we present several examples of the computation of kappa using both fourier analysis and standard finite element techniques we compare the predictions of our theorems with the actual rate of convergence our analysis also shows that

error estimates for a finite volume element method for elliptic pdes in nonconvex polygonal domains we consider standard finite volume piecewise linear approximations for second order elliptic boundary value problems on a nonconvex polygonal domain based on sharp shift estimates we derive error estimations in h1 l2 and l infty norms taking into consideration the regularity of the data numerical experiments and counterexamples illustrate the theoretical results

on the sharpness of certain local estimates for projections into finite element spaces influence of a re entrant corner in a plane polygonal domain with a reentrant corner consider a homogeneous dirichlet problem for poisson s equation au f with f smooth and the corresponding galerkin finite element solutions in a family of piecewise polynomial spaces based on quasi uniform uniformly regular triangulations with the diameter of each element compara

on generalized powers stormer s inequality abstract a generalization of powers stormer s inequality for operator monotone functions on 0 and for positive linear functional on general c algebras will be proved it also will be shown that the generalized powers stormer inequality characterizes the tracial functionals on c algebras

new estimates for multilevel algorithms including the v cycle the purpose of this paper is to provide new estimates for certain multilevel algorithms in particular we are concerned with the simple additive multilevel algorithm discussed recently together with j xu and the standard v cycle algorithm with one smoothing step per grid we shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels even on nonconvex domains which do not provide full elliptic regularity for example the theory applies to the standard multigrid vcycle on the l shaped domain or a domain with a crack and yields a uniform convergence rate we also prove uniform convergence rates for the multigrid v cycle for problems with nonuniformly refined meshes finally we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid v cycle algorithms

the post processing approach in the finite element method part 2 the calculation of stress intensity factors abstract in the context of a model problem the authors describe post processing techniques for the calculation of generalized stress intensity factors they discuss two broad classes of methods one involving an influence function and the other related to the well known energy release principle of fracture mechanics an error analysis is sketched and two numerical examples are given to illustrate the effectivity of the techniques this is the second in a series of three papers author

a numerical procedure for the determination of certain quantities related to the stress intensity factors in two dimensional elasticity abstract this work is devoted to the investigation of corner and interface singularities in linear elasticity and to the development of a numerical code that computes certain quantities associated with those singularities we compute several terms of the asymptotic expansion of the solution of the elasticity system near the corner or interface which is of the form knr nfn in particular the n and fn we treat our problem as a non linear eigenvalue problem and we use an approach based on the shooting method for boundary value problems to create an analytic function whose zeros are the n above the zeros of this function are then found using an iteration method and a global method to make sure that no zero was overlooked the program works for the general case of anisotropic non homogeneous wedge with general boundary conditions as well as in the case of crossing interfaces in an internal point we test the convergence accuracy robustness efficiency and reliability of the code from those tests it can be seen that the code works quite well for the majority of the applications

on the determination of higher order terms of singular elastic stress fields near corners in two dimensional elasticity stresses at reentrant corners exhibit singular behavior the stress field is of the form sigma sigma rm k iota r lambda 2 1 f i theta lambda i where r are polar coordinates centered at the tip of the corner andf i i are smooth functions for practical use of this series the eigenvalues i which are generally complex numbers are required in order of ascending real part the problem then is to find the roots of a transcendental equation eigenequation in the complex plane and arranged in order of ascending real part a theorem is proved on the number location and nature of the roots of this equation with the real part in fixed intervals of length excellent initial estimates of the real part of the complex roots become available and so are bounds within which single real roots exist this enables the determination of any number of roots in ascending order of real part the critical angles at which the eigenvalues change nature are also determined it is shown that for certain cases and for the symmetric mode of deformation the eigenvalue 1 does not represent a rigid body rotation therefore it has to be included in the series representation of the stresses the coefficientsk i can be determined by recently developed extraction techniques thus allowing complete determination of the elastic field and enabling its correlation with experimental data on brittle fracture crack initiation plastic zone estimation etc

block method for problems on l shaped domains an extremely accurate exponentially convergent solution is presented for both symmetric and non symmetric laplacian problems on l shaped domains by using one block version of the block method bm a simple and highly accurate formula for computing the stress intensity factor is given comparisons with various results in the literature are included

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

a finite element method using singular functions for poisson equations mixed boundary conditions in z cai s kim a finite element method using singular functions for the poisson equation corner singularities siam j numer anal 39 2001 286 299 we proposed a new finite element method to compute singular solutions of poisson equations on a polygonal domain with re entrant angles singularities are eliminated and only the regular part of the solution that is in h 2 is computed the stress intensity factor and the solution can be computed as a post processing step this method is extended to problems with crack singularities and to a higher order method for smooth data in z cai s kim g woo a finite element method using singular functions for the poisson equation crack singularities numer linear algebra appl 9 2002 445 455 in this paper we study the poisson equation with mixed boundary conditions examples with various singular points and numerical results are presented 2005 elsevier b v all rights reserved

successive overrelaxation sor and related methods abstract covering the last half of the 20th century we present some of the basic and well known results for the sor theory and related methods as well as some that are not as well known most of the earlier results can be found in the excellent books by varga matrix iterative analysis prentice hall englewood cliffs nj 1962 young iterative solution of large linear systems academic press new york 1971 and berman and plemmons nonnegative matrices in the mathematical sciences siam philadelphia pa 1994 while some of the most recent ones are given in the bibliography of this paper in this survey both the point and the block sor methods are considered for the solution of a linear system of the form ax b where a c n n and b c n 0 some general results concerning the sor and related methods are given and also some more specific ones in cases where a happens to possess some further property e g positive definiteness l m h matrix property p cyclic consistently ordered property etc

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

overcoming corner singularities using multigrid methods we consider the poisson equation delta u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain o we develop a finite element multigrid method on quasi uniform grids that obtains cal o h m 1 epsilon convergence in the h 1 o norm for any positive epsilon when f in h m o the cost of the method is proportional to the number of elements in the triangulation the results of this paper can be generalized to other equations and other boundary conditions

multigrid methods for variational problems further results in an earlier paper siam j numer anal 19 1982 pp 924 929 we developed a convergence theory for a class of multigrid methods applied to positive definite self adjoint differential boundary value problems where the multigrid process used w cycling and richardson s iteration in the present paper we extend this theory to include v cycling and more general relaxation schemes such as jacobi and arbitrarily ordered gauss seidel

a singular function boundary integral method for laplacian problems with boundary singularities a singular function boundary integral method for laplacian problems with boundary singularities is analyzed in this method the solution is approximated by the truncated asymptotic expansion for the solution near the singular point and the dirichlet boundary conditions are weakly enforced by means of lagrange multiplier functions the resulting discrete problem is posed and solved on the boundary of the domain away from the point of singularity the main result of this paper is the proof of convergence of the method in particular we show that the method approximates the generalized stress intensity factors i e the coefficients in the asymptotic expansion at an exponential rate a numerical example illustrating the convergence of the method is also presented