overlapping schwarz methods for mixed linear elasticity and stokes problems overlapping schwarz preconditioners are introduced and studied for saddle point problems with a penalty term such as stokes equations and mixed formulations of linear elasticity these preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem numerical experiments show that these are parallel and scalable preconditioners since the rate of convergence of the preconditioned operator is independent of the mesh size h the number of subdomains n and the penalty parameter

the locally stabilized finite volume method for incompressible flow a locally stabilized finite volume method for the two dimensional incompressible flow governed by stationary stokes and navier stokes equations is investigated in this work a macroelement condition is introduced for constructing the local stabilized formulation for the problem the optimal l 2 error estimate of u h p h determined by stokes equations is also deduced some numerical examples are provided to check the efficiency of this method

analysis of mixed finite elements methods for the stokes problem a unified approach we develop a method for the analysis of mixed finite element methods for the stokes problem in the velocity pressure formulation a technical macroelement condition which is sufficient for the classical babuska brezzi inequality to be valid is introduced using this condition we are able to verify the stability and optimal order of convergence of several known mixed finite element methods

ellipsoids that contain all the solutions of a positive semi definite linear complementarity problem this paper deals with the lcp linear complementarity problem with a positive semi definite matrix assuming that a strictly positive feasible solution of the lcp is available we propose ellipsoids each of which contains all the solutions of the lcp we use such an ellipsoid for computing a lower bound and an upper bound for each coordinate of the solutions of the lcp we can apply the lower bound to test whether a given variable is positive over the solution set of the lcp that is if the lower bound is positive we know that the variable is positive over the solution set of the lcp hence by the complementarity condition its complement is zero in this case we can eliminate the variable and its complement from the lcp we also show how we efficiently combine the ellipsoid method for computing bounds for the solution set with the path following algorithm proposed by the authors for the lcp if the lcp has a unique non degenerate solution the lower bound and the upper bound for the solution computed at each iteration of the path following algorithm both converge to the solution of the lcp

stabilized discontinuous finite element approximations for stokes equations in this paper we derive two stabilized discontinuous finite element formulations symmetric and nonsymmetric for the stokes equations and the equations of the linear elasticity for almost incompressible materials these methods are derived via stabilization of a saddle point system where the continuity of the normal and tangential components of the velocity displacements are imposed in a weak sense via lagrange multipliers for both methods almost all reasonable pair of discontinuous finite element spaces can be used to approximate the velocity and the pressure optimal error estimate for the approximation of both the velocity of the symmetric formulation and pressure in l2 norm are obtained as well as one in a mesh dependent norm for the velocity in both symmetric and nonsymmetric formulations

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

existence of solutions to the self dual equations in the maxwell gauged o 3 sigma model abstract in this paper we prove the existence of solutions in the self dual maxwell gauged o 3 sigma model by the super and subsolution method the supersolutions are explicitly constructed and the subsolutions are given in different ways according to the parameter representing the decay rates of solutions at infinity when has noncritical values we obtain subsolutions by solving laplace equations on weighted sobolev spaces if is the critical number we find the subsolutions via induction and shooting arguments

a multigrid version of a simple finite element method for the stokes problem we consider a finite element method for the stokes problem on a rectangular domain based on piecewise bilinear velocities and piecewise constant pressures on a uniform rectangular grid it is shown that by a simple stabilization strategy the method can be implemented in a convergent multigrid procedure introduction one of the most natural ways of discretizing the stokes equations on a two dimensional rectangular domain is to apply the finite element techniques with continuous piecewise bilinear velocities and piecewise constant pressures on a rectangular grid this method is easily extended to more general quadrilateral meshes and it has proved to be quite effective in practice the convergence analysis of the method was first carried out in 10 on a rectangular domain in 12 the estimates of 10 are improved and extended to more general quadrilateral meshes so far the above method has been used mainly in connection with direct band solvers usually combined with penalty perturbation techniques to eliminate the pressure 9 in this paper we consider an iterative variant of the method based on multigrid techniques such an algorithm seems attractive especially since the un derlying finite difference equations are relatively simple multigrid methods for the stokes problem and for more general elliptic systems were considered previously by hackbusch 8 and verfurth 14 in 14 it is shown that a convergent multigrid algorithm for the stokes problem can be constructed by appropriately scaling the variables provided that the underlying finite element scheme satisfies the babu ka brezzi stability condition 2 5 in our case however the finite element method is not stable in this sense and we have not been able to show that the straightforward application of the algorithm proposed in 14 yields a convergent process therefore we suggest first modifying the finite element method in such a way that it becomes stable in the ordinary sense the usual way of stabilizing an unstable mixed method is to add more velocity or primary variables a classical example of this is the quadratic linear velocity pres sure element of crouzeix and raviart 6 where added bubble functions act as stabilizers in the velocity space increasing the dimension of the velocity space would be an alternative in our case however we propose another method which seems to yield a simpler algorithm it is based on adding an extra stabilizing term into the

a multilevel algorithm for mixed problems we describe a multilevel algorithm for the numerical solution of symmetric indefinite problems which arise e g from mixed finite element approximations of the stokes equation the main difficulty besides the indefiniteness is the lack of regularity of the solution of the corresponding continuous problem this is overcome by introducing a scale of mesh dependent norms the convergence rate of the described algorithm is bounded independently of the meshsize for convenience we only discuss jacobi relaxation as smoothing operator in detail in the last section we comment on lanczos type smoothing procedures

analysis of locally stabilized mixed finite element methods for the stokes problem in this paper a locally stabilized finite element formulation of the stokes problem is analyzed a macroelement condition which is sufficient for the stability of locally stabilized mixed methods based on a piecewise constant pressure approximation is introduced by satisfying this condition the stability of the qi po quadrilateral and the pi po triangular element can be established

stability of finite elements under divergence constraints a finite dimensional stability test for checking velocity pressure finite element trial spaces is presented applications are made to a new class of element pairs proposed in this paper as well as to existing spaces

stabilization of low order cross grid pkql mixed finite elements abstract in this paper we analyze a low order family of mixed finite element methods for the numerical solution of the stokes problem and a second order elliptic problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals for the lowest order p 1 q 0 a global spurious pressure mode is shown to exist and so this element as p 1 q 1 case analyzed in armentano and blasco 2010 is unstable however following the ideas given in bochev et al 2006 a simple stabilization procedure can be applied when we approximate the solution of the stokes problem such that the new p 1 q 0 and p 1 q 1 methods are unconditionally stable and achieve optimal accuracy with respect to solution regularity with simple and straightforward implementations moreover we analyze the application of our p 1 q 1 element to the mixed formulation of the elliptic problem in this case by introducing the modified mixed weak form proposed in brezzi et al 1993 optimal order of accuracy can be obtained with our stabilized p 1 q 1 elements numerical results are also presented which confirm the existence of the spurious pressure mode for the p 1 q 0 element and the excellent stability and accuracy of the new stabilized methods

analysis of the cell vertex finite volume method for hyperbolic problems with variable coefficients an analysis of the cell vertex finite volume method is presented for a scalar linear hyperbolic equation with variable coefficients the cell vertex approximation is shown to be second order convergent in the l2 norm and first order convergent in the h1 norm provided the exact solution belongs to h3 and the characteristic curves of the differential equation are transversal to mesh lines on the closure of the domain of integration when characteristics are parallel to mesh lines on a subset of positive measure the scheme is only first order convergent in the l2 norm however if transversality fails on part of the outflow boundary only then the cell vertex scheme is second order convergent in a weighted l2 norm

optimal spectral approximation of 2n order differential operators by mixed isogeometric analysis abstract we approximate the spectra of a class of 2 n order differential operators using isogeometric analysis in mixed formulations this class includes a wide range of differential operators such as those arising in elliptic biharmonic cahn hilliard swift hohenberg and phase field crystal equations the spectra of the differential operators are approximated by solving differential eigenvalue problems in mixed formulations which require auxiliary parameters the mixed isogeometric formulation when applying classical quadrature rules leads to an eigenvalue error convergence of order 2 p where p is the order of the underlying b spline space we improve this order to be 2 p 2 by applying optimally blended quadrature rules developed in puzyrev et al 2017 caloet al 0000 and this order is an optimum in the view of dispersion error we also compare these results with the mixed finite elements and show numerically that the mixed isogeometric analysis leads to significantly better spectral approximations

optimal error estimate of the penalty finite element method for the time dependent navier stokes equations a fully discrete penalty finite element method is presented for the two dimensional time dependent navier stokes equations the time discretization of the penalty navier stokes equations is based on the backward euler scheme the spatial discretization of the time discretized penalty navier stokes equations is based on a finite element space pair x h m h which satisfies some approximate assumption an optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters e at and h are sufficiently small

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

a stabilized nonconfirming finite element method based on multiscale enrichment for the stationary navier stokes equations abstract in the paper we proposed a stabilized nonconforming finite element method for the stationary incompressible navier stokes equations which is a petrov galerkin approach based on the enrichment of the standard polynomial space for the velocity component with multiscale functions and we studied the stability of the method for p 1 p 0 triangular element q 1 p 0 quadrilateral element and obtained the optimal error estimates of the stabilized nonconforming finite element method for the stationary navier stokes equations

corrigenda the convergence of accelerated overrelaxation iterations math comp 47 1986 no 175 219 223 mr0842131 87g 65046

stokes coupling method for the exterior flow part iv stabilized finite element approximation based on the stokes coupling method for solving the two dimensional exterior unsteady navier stokes equations the stabilized finite element approximation combining the boundary element with the stabilized finite element for solving the reduced stokes coupling equations is presented finally the optimal error estimates for the finite element solution are derived

stable nonconforming quadrilateral finite elements for the stokes problem stability result is obtained for the approximation of the stationary stokes problem with nonconforming elements proposed by douglas et al j douglas jr j e santos d s xiu ye nonconforming galerkin methods based on quadrilateral elements for second order elliptic problems preprints added by conforming bubbles to the velocity and discontinuous piecewise linear to the pressure on quadrilateral elements optimal order error estimates derived