efficient block preconditioning for a c1 finite element discretisation of the dirichlet biharmonic problem we present an efficient block preconditioner for the two dimensional biharmonic dirichlet problem discretised by c1 bicubic hermite finite elements in this formulation each node in the mesh has four different degrees of freedom dofs grouping dofs of the same type together leads to a natural blocking of the galerkin coefficient matrix based on this block structure we develop two preconditioners a 2x2 block diagonal preconditioner bd and a block bordered diagonal bbd preconditioner we prove mesh independent bounds for the spectra of the bd preconditioned galerkin matrix under certain conditions the eigenvalue analysis is based on the fact that the proposed preconditioner like the coefficient matrix itself is symmetric positive definite and is assembled from element matrices we demonstrate the effectiveness of an inexact version of the bbd preconditioner which exhibits near optimal scaling in terms of computational cost with respect to the discrete problem size finally we study robustness of this preconditioner with respect to element stretching domain distortion and non convex domains

multigrid preconditioning for the biharmonic dirichlet problem a multigrid preconditioning scheme for solving the ciarlet raviart mixed method equations for the biharmonic dirichlet problem is presented in particular a schur complement formulation for these equations which yields noninherited quadratic forms is considered the preconditioning scheme is compared with a standard w cycle multigrid iteration it is proved that a variable v cycle preconditioner leads to problems with uniformly bounded condition numbers however w cycle convergence is proved only if the number of smoothings m is sufficiently large an example is given in which the w cycle diverges unless m geq 8 divergent w cycles are also encountered when solving the morley equations for the biharmonic dirichlet problem although brenner has proved w cycle convergence for sufficiently large m 9 this is illustrated with additional computations while variable v cycles continue to produce excellent preconditioners in this setting certain approximate l 2 inner products are described and a modif

a multigrid method for the mortar type morley element approximation of a plate bending problem in this paper a multigrid algorithm is presented for the mortar element method for the morley nonconforming plate element based on the multigrid theory developed by bramble pasciak and xu in math comp 56 1991 pp 1 34 we prove that the w cycle multigrid method is optimal i e the convergence rate is independent of the mesh size and mesh level moreover a variable v cycle multigrid preconditioner is constructed which results in a preconditioned system with a uniformly bounded condition number

the poor bending response of the four node plane stress quadrilateral many plane stress finite elements which can exactly represent rigid body and constant strain modes are too stiff in their response to the simple flexural action of a beam this problem has received considerable interest this paper explores a new interpretation of the problem and shows that the poor bending response of the original 4 node plane stress quadrilateral can be quantitatively predicted by an error model

convergence of nonconforming multigrid methods without full elliptic regularity we consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity we prove that there is a bound 1 for the contraction number of the w cycle algorithm which is independent of mesh level provided that the number of smoothing steps is sufficiently large we also show that the symmetric variable v cycle algorithm is an optimal preconditioner

a v cycle multigrid method and additive multilevel preconditioners for the plate bending problem discretized by conforming finite elements it is well known that many conforming plate finite element spaces in the multilevel methods are nonnested even when the triangulations are nested so it is difficult to construct the effective v cycle multigrid method and multilevel preconditioners for the plate bending problem in this paper a new method dealing with multilevel methods for finite element discretisation equations of the plate problem is developed an optimal v cycle multigrid method and multilevel preconditioners for some famous conforming plate elements are constructed moreover our methods can be applied to the plate problem with complicated geometries

hierarchical bases for elliptic problems linear systems of equations with positive and symmetric matrices often occur in the numerical treatment of linear and nonlinear elliptic boundary value problems if the cg algorithm is used to solve these equations one is able to speed up the convergence by preconditioning the method of preconditioning with hierarchical basis has already been considered for the laplace equation in two space dimensions and for linear conforming elements in the present work this method is generalized to a large class of conforming and nonconforming elements 0 introduction consider elliptic problems in variational form of order 2m let x htm 2 ci and q w be a bounded domain we want to find u x such that 0 1 a u v i a vmu vmv f v vv x ja here a denotes a strictly positive possibly x dependent matrix f is a functional on x and a is assumed to satisfy c0lmlm 2 o colmlm 2 0 for all u x the solution of this problem exists according to the lax milgram theorem one way to solve this problem numerically is to approximate x by a sequence of finite dimensional spaces xn for a en if ui i xyn is a basis of xn writing u in the form

bpx type preconditioners for second and fourth order elliptic problems on the sphere we develop two bramble pasciak xu type preconditioners for second resp fourth order elliptic problems on the surface of the two sphere to discretize the second order problem we construct c 0 linear elements on the sphere and for the fourth order problem we construct c 1 finite elements of powell sabin type on the sphere the main reason these bpx preconditioners work depends on this particular choice of basis we prove optimality and provide numerical examples furthermore we numerically compare the bpx preconditioners with the suboptimal hierarchical basis preconditioners

the hierarchical basis multigrid method we derive and analyze the hierarchical basis multigrid method for solving discretizations of self adjoint elliptic boundary value problems using piecewise linear triangular finite elements the method is analyzed as a block symmetric gau seidel iteration with inner iterations but it is strongly related to 2 level methods to the standard multigridv cycle and to earlier jacobi like hierarchical basis methods the method is very robust and has a nearly optimal convergence rate and work estimate it is especially well suited to difficult problems with rough solutions discretized using highly nonuniform adaptively refined meshes

a v cycle multigrid method for trunc plate element in this paper an optimal v cycle multigrid method is presented for the trunc plate element the uniform convergence rate independent of mesh sizes and levels is established the method is extended to bergan s energy orthogonal plate element

determinism and non determinism in pdl abstract the main purpose of this paper is to consider a variant of propositional dynamic logic pdl in which the natural program concepts while do od and if then else fi are basic and in which is the only non deterministic program concept dropping leads to the known system strict propositional dynamic logic spdl the main result of this paper is that spdl is weaker than pdl it is pointed out however that by extending spdl by propositional assignments a system called propositional program logic ppl is obtained which has the same expressive power as pdl

preconditioners for nonconforming discretizations we prove an abstract norm equivalence for a two level method which allows us to reduce the construction of preconditioners for nonconforming finite element discretizations to known cases of conforming elements

algebraic study of multigrid methods for symmetric definite problems abstract we present a convergence theory based on a simple algebraic assumption which permits us to analyze multilevel iterative methods for symmetric positive definite linear systems coarse grid problems are derived variationally we prove fast convergence of v and w cycles with any positive number of smoothing steps under a discrete analogue of the h 2 and h 1 ga regularity assumption respectively and for a wide class of smoothings including arbitrarily ordered gauss seidel and steepest descent

c 1 hierarchical bases abstract in this note we derive estimates for the condition numbers of stiffness matrices relative to certain c1 conforming hierarchical bases for fourth order elliptic boundary value problems

multilevel schwarz methods for the biharmonic dirichlet problem the author considers the solution of the algebraic system of equations that result from the finite element discretization of the biharmonic equation some multilevel algorithms are designed and analyzed using a schwarz framework both additive and multiplicative variants of the algorithms are considered and condition number estimates for the additive algorithms and the energy norm estimates for the error propagation operator of the multiplicative algorithms are given it is noted that for a proper ordering the iterative operators of the multiplicative algorithms correspond to the error propagation operators of certain v cycle multigrid methods

extension theorems for plate elements with applications extension theorems for plate elements are established their applications to the analysis of nonoverlapping domain decomposition methods for solving the plate bending problems are presented numerical results support our theory

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

linear stability analysis for the multikink solution of the sine hilbert equation the linear stability analysis is performed for the multikink solution of the sine hilbert sh equation we first solve the eigenvalue problem of the sh equation linearized about the multikink solution and then establish the completeness relation among the eigenfunctions this relation enables us to solve the initial value problem of the linearized sh equation proving the stability of the solution against infinitesimal perturbations as an application of the completeness relation we develop a direct multikink perturbation theory while employing the method of multiple time scales we derive the evolution equations that describe the slow change of the kink parameters as well as a nonsecular expression for the first order correction to the multikink solution a novelty of the present analysis is that the proof of the completeness relation is done by purely algebraic means with the help of the classical theory of algebraic equations

c 1 spline wavelets on triangulations in this paper we investigate spline wavelets on general triangulations in particular we are interested in c 1 wavelets generated from piecewise quadratic polynomials by using the powell sabin elements we set up a nested family of spaces of c 1 quadratic splines which are suitable for multiresolution analysis of besov spaces consequently we construct c 1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three direction mesh a general theory is developed so as to verify the global stability of these wavelets in besov spaces the wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations

a fast solver for the orthogonal spline collocation solution of the biharmonic dirichlet problem on rectangles a fast schur complement algorithm is presented for computing the piecewise hermite bicubic orthogonal spline collocation solution of the biharmonic dirichlet problem on a rectangular region on an n n uniform partition the algorithm which involves the preconditioned conjugate gradient method and fast fourier transforms requires o n2 log2 n arithmetic operations

on v cycle multigrid implementation for plate elements in this paper an optimal v cycle multigrid method is presented for the nonconforming element approximation of the plate bending problem does not need full elliptic regularity the uniform convergence rate independent of mesh sizes and levels is established some numerical tests are given to illustrate the efficiency of the method

the propositional dynamic logic of deterministic well structured programs abstract we consider a restricted propositional dynamic logic strict deterministic propositional dynamic logic sdpdl which is appropriate for reasoning about deterministic well structured programs in contrast to pdl for which the validity problem is known to be complete in deterministic exponential time the validity problem for sdpdl is shown to be polynomial space complete we also show that sdpdl is less expressive than pdl and give a complete axiomatization for it the results rely on structure theorems for models of satisfiable sdpdl formulas and the proofs give insight into the effects of nondeterminism on intractability and expressiveness in program logics

thermomechanical response of solids at high strains natural approach abstract the paper develops in pursuance of 4 the theoretical framework appertaining to coupled thermomechanical deformations of solids subject to large as well as inelastic deformations the essential feature of the analysis is a consistent natural formulation which encompasses also all thermodynamic aspects on the other hand it must be admitted that current concepts on mechanical constitutive assumptions 5 6 are not sufficiently sophisticated to permit a precise generalisation for the purpose of incorporating thermomechanical effects and delivering a fully compatible theory of the coupled problem uniqueness and bifurcation of the solutions as well as stable postcritical deformation paths are also considered in this account furthermore algorithms for the solution of the nonlinear algebraic equations representing the discretised problem and in particular quasi newton techniques are discussed the paper concludes with a number of applications of technical relevance

refinable c2 piecewise quintic polynomials on powell sabin 12 triangulations we present a construction of nested spaces of c 2 macro elements of degree 5 on triangulations of a polygonal domain obtained by uniform refinements of an initial triangulation and a powell sabin 12 split

on multi level bases for elliptic boundary value problems abstract we study the multi level method for preconditioning a linear system arising from a galerkin discretization method of an elliptic boundary value problem of order 2r the solution is approximated in the spline space s10 n when r 1 and s3r 1r 1 n or s 3r 3 r 1 n when r 2 where sd denotes a spline space of smoothness and degree d n is the nth either uniform or nonuniform refinement of a given triangulation 0 and n is the triangulation obtained from the nth refinement either uniform or nonuniform of a given quadrangulation we show that we can construct a multi level basis in these spline spaces which preconditions the linear system so that its condition number is o n 1 2

dynamics of solitons in a damped sine hilbert equation a damped sine hilbert sh equation is proposed it can be linearized by a dependent variable transformation which enables one to solve an initial value problem of the equation the n soliton solution is obtained explicitly and its properties are investigated in comparison with those of the n soliton solution of the sh equation in particular the interaction of the two solitons is explored in detail with the aid of the pole representation it is found that the interaction process is classified into the two types according to the initial amplitudes and positions of both solitons in the general n soliton case the long time behavior of the solution is shown to be characterized by the positive n zeros of the hermite polynomial of degree 2n finally a linearized version of the damped sh equation is briefly discussed

inexact hierarchical scale separation a two scale approach for linear systems from discontinuous galerkin discretizations abstract hierarchical scale separation hss is an iterative two scale approximation method for large sparse systems of linear equations arising from discontinuous galerkin dg discretizations hss splits the linear system into a coarse scale system of reduced size corresponding to the local mean values of the solution and a set of decoupled local fine scale systems corresponding to the higher order solution components this scheme then alternates between coarse scale and fine scale system solves until both components converge the motivation of hss is to promote parallelism by decoupling the fine scale systems and to reduce the communication overhead from classical linear solvers by only applying them to the coarse scale system we propose a modified hss scheme inexact hss ihss that exploits the highly parallel fine scale solver more extensively and only approximates the coarse scale solution in every iteration thus resulting in a significant speedup the tolerance of the coarse scale solver is adapted in every ihss cycle controlled by the residual norm of the fine scale system anderson acceleration is employed in the repeated solving of the fine scale system to stabilize the scheme we investigate the applicability of ihss to systems stemming from the nonsymmetric interior penalty dg discretization of the cahn hilliard equation discuss its hybrid parallel implementation for large scale simulations and compare the performance of a widely used iterative solver with and without ihss

an additional stiffness parameter measure of error of the second kind in the finite element method conventional error analyses norms in the finite element method are based on the percentage error or its equivalent in some computed value as compared to the theoretically predicted value in problems where two or more field variables are coupled it is possible that if the field interpolation functions do not consistently reproduce limiting situations of physical behaviour there may be very large errors which are exaggerated enormously when some structural parameters become indefinitely large in a penalty limiting sense the percentage error norms therefore saturate quickly to a value approaching 100 per cent and do not sensibly reflect the relationship between error and the structural parameter even on a logarithmic plot errors due to these spurious constraints have recently been recognized as belonging to a special class described as errors of the second kind a new error norm called the additional stiffening parameter helps to recognize the manner in which errors of the second kind can be blown out of propprtion by a large variation in some structural parameter recent investigations in some multi field problems are summarized here to demonstrate the usefulness of this concept

on the initial value problem for a class of nonlinear integral evolution equations including the sine hilbert equation a method for solving a class of nonlinear singular integral evolution equations for decaying initial values on the line is presented the underlying scattering problem is a matrix riemann hilbert problem scattering analysis shows that the spectrum is purely discrete an application is to the so called sine hilbert equation h t c sin where c is a constant and h denotes the hilbert transform

an optimal order nonconforming multigrid method for the biharmonic equation an optimal order multigrid method for solving the biharmonic equation using morley nonconforming finite elements is developed

bivariate c 1 quadratic finite elements and vertex splines following work of heindl and of powell and sabin each triangle of an arbitrary regular triangulation a of a polygonal region ii in r is subdi vided into twelve triangles using the three medians yielding the refinement a of a so that c quadratic finite elements can be constructed in this paper we derive the bezier nets of these elements in terms of the parameters that de scribe function and first partial derivative values at the vertices and values of the normal derivatives at the midpoints of the edges of a consequently bivariate c quadratic generalized vertex splines on a have an explicit formulation here a generalized vertex spline is one which is a piecewise polynomial on the refined grid partition a and has support that contains at most one vertex of the original partition a in its interior the collection of all c quadratic general ized vertex splines on a so constructed is shown to form a basis of s2 a the vector space of all functions on c i2 whose restrictions to each triangular cell of the partition a are quadratic polynomials a subspace with the basis given by appropriately chosen generalized vertex splines with exactly one vertex of a in the interior of their supports that reproduces all quadratic polynomials is identified and hence has approximation order three quasi interpolation for mulas using this subspace are obtained in addition a constructive procedure that yields a locally supported basis of yet another subspace with dimension given by the number of vertices of a that has approximation order three is given

direct and inverse scattering problems of the nonlinear intermediate long wave equation the inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed the equation describes long internal gravity waves in a stratified fluid of finite depth and reduces to the korteweg de vries equation as shallow water limit and the benjamin ono equation as deep water limit both limits of the method and novel aspects of the theory are also discussed

interpolation of besov spaces we investigate besov spaces and their connection with dyadic spline approximation in lp q 0 p oo our main results are the determination of the interpolation spaces between a pair of besov spaces an atomic decomposition for functions in a besov space the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines

on the limit from the intermediate long wave equation to the benjamin ono equation the intermediate long wave equation is a physically important singular integrodifferential equation containing a parameter referred to here as for it reduces to the benjamin ono equation it has been recently shown that the inverse scattering transform schemes of the above equations have certain significant differences here it is shown that for the inverse scattering transform scheme of the intermediate long wave equation reduces to that of the benjamin ono equation

trunc for shells an element possibly to the taste of bruce irons in his delightful and original textbook 18 bruce irons introduces the chapter of plate bending with the remark history of disasters but the pioneering days are past this is certainly true on the other hand we learn in the same context from irons we believe that the world has not yet exhausted the older formulations in this paper we present a review of theoretical aspects of the plate bending triangle trunc reference 4 together with extensive tests in linear statics dynamics buckling and nonlinear large deformations both for plates and shells the unconventional trunc model is based on the patch test which was originated by irons 1 in spite of its limited theoretical foundation the performance of this element on a fairly broad field of applications appears at least from the practical point of view very satisfactory

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

a two level additive schwarz preconditioner for nonconforming plate elements a two level additive schwarz preconditioner is developed for the systems resulting from the discretizations of the plate bending problem by the morley finite element the fraeijs de veubeke finite element the zienkiewicz finite element and the adini finite element the condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of a generous overlap

hierarchical conforming finite element methods for the biharmonic equation the paper deals with hierarchical bases in spaces of conforming c 1 elements in connection with the approximate solution of the biharmonic equation delta 2 u f quad text in omega qquad u frac partial u partial n 0 quad text on partial omega on a plane polygonal domain omega two different composite finite elements are studied piecewise quadratic powell sabin elements and piecewise cubic elements of clough tocher type the main result are estimates for the condition numbers of the corresponding discretization matrices that show that a conjugate gradient method applied to the hierarchical discretization the so called hierarchical multilevel method will yield suboptimal convergence rates in comparison with standard multigrid schemes

some considerations on the natural approach abstract in this note we present some pertinent comments on the natural mode technique in the first part we demonstrate that for arbitrary finite elements the choice of urelements may exclusively be based on simplex elements in fact the corresponding infinitesimal building blocks the so called subelements may be used also for quadrilateral and hexahedronal elements or for that matter for any one two or three dimensional element the second part develops some new considerations on unconventional models associated with the patch test

a note on convergence of v cycle nonconforming multigrid methods1this research was supported by the doctoral program foundation of china universities and by nsf and china postdoctoral science foundation 1 in this paper the nonconforming multigrid methods are studied i e the fine mesh spaces and coarse mesh spaces also use the same nonconforming finite element contrary to the observation by chen convergence of multigrid method for nonconforming method pre print 1996 even if quadratic forms are not changed on coarse meshes v cycle nonconforming multigrid methods still can converge the key of our methods is the choice of a good intergrid operator moreover optimal convergent order of our v cycle nonconforming multigrid methods is obtained

propositional dynamic logic with local assignment we propose an extension of propositional dynamic logic which allows a new kind of program terms local assignments to propositional variables they are very close to the known array assignments in dynamic logic beacuse they allow us to change the truth values of predicate variables in this logic many notions like equivalence of programs looping and finitely branching are expressible on a propositional level in fact we show that the resulting logic is equivalent in expressive power to first order logic augmented by a device to express transitive closures in other words it is modulo extra predicate symbols equivalent to first order dynamic logic not suprisingly therefore the validity problem for this extension is 11 complete

an analysis of the trunc element for coupled plates with rigid junction abstract this paper is devoted to the numerical analysis of the trunc plate element for coupled plates meeting at an arbitrary angle in which different boundary conditions such as clamped natural boundary condition and rigid junction condition may appear on different parts of the boundary of coupled plates the convergence of the trunc element for such a problem is established and the error estimate is derived

legendre galerkin spectral element method for the biharmonic equations and its applications abstract a spectral element method based on the legendre galerkin approximation is presented to solve the two dimensional biharmonic equations rigorous error analysis is carried out to establish the convergence of the method by constructing appropriate basis functions which satisfy the boundary conditions of the differential equations the discrete variational formulation is reduced to linear system with sparse and symmetric matrices which can be efficiently solved by a fast schur complement method accuracy test is provided to confirm the convergence rate of the theoretical results finally the proposed method is applied to calculate the displacement of an elastic plate under a uniform applied load and stream function of zero reynolds number flow in a driven cavity the results compare well with established results

convergence of nonconforming v cycle and f cycle multigrid algorithms for second order elliptic boundary value problems the convergence of v cycle and f cycle multigrid algorithms with a sufficiently large number of smoothing steps is established for nonconforming finite element methods for second order elliptic boundary value problems

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

robust multigrid solvers for the biharmonic problem in isogeometric analysis abstract in this paper we develop multigrid solvers for the biharmonic problem in the framework of isogeometric analysis iga in this framework one typically sets up b splines on the unit square or cube and transforms them to the domain of interest by a global smooth geometry function with this approach it is feasible to set up h 2 conforming discretizations we propose two multigrid methods for such a discretization one based on gauss seidel smoothing and one based on mass smoothing we prove that both are robust in the grid size the latter is also robust in the spline degree numerical experiments illustrate the convergence theory and indicate the efficiency of the proposed multigrid approaches particularly of a hybrid approach combining both smoothers

a finite element method for investigating general elastic multi structures a finite element method is proposed for investigating the general elastic multi structure problem where displacements on bodies longitudinal displacements on plates longitudinal displacements and rotational angles on rods are discretized using conforming linear elements transverse displacements on plates and rods are discretized respectively using trunc elements and hermite elements of third order and the discrete generalized displacement fields in individual elastic members are coupled together by some feasible interface conditions the unique solvability of the method is verified by the lax milgram lemma after deriving generalized korn s inequalities in some nonconforming element spaces on elastic multi structures the quasi optimal error estimate in the energy norm is also established some numerical results are presented at the end

a conjugate gradient method and a multigrid algorithm for morley s finite element approximation of the biharmonic equation the numerical solution of the linear equations arising from morley s nonconforming displacement method is studied a suitable preconditioning for the conjugate gradient method is described furthermore the nonconformity of the discretization necessitates some non standard ingredients of multigrid algorithms

a least squares method for solving biharmonic problems we discuss a generalization of the cauchy riemann equations which is applicable to biharmonic problems a first order system is derived from these equations and this system is solved using the least squares method with continuous trial functions set up by the finite element method optimal rates of convergence in l 2 omega are proved in regions with smooth boundaries and optimal rates of convergence are proved in h 1 omega and using linear triangular elements observed in l 2 omega in polygonal regions

a multilevel algorithm for the biharmonic problem a finite element discretization of the mixed variable formulation of the biharmonic problem is considered a multilevel algorithm for the numerical solution of the discrete equations is described convergence is proved under the assumption ofh 3 regularity

an optimal order multigrid method for p1 nonconforming finite elements etude d une methode multigrille d ordre optimal pour la resolution des problemes aux limites elliptiques du second ordre a l aide d elements finis non conformes

numerical solution of partial differential equations with powell sabin splines powell sabin splines are piecewise quadratic polynomials with global c 1 continuity they are defined on conforming triangulations of two dimensional domains and admit a compact representation in a normalized b spline basis recently these splines have been used successfully in the area of computer aided geometric design for the modelling and fitting of surfaces in this paper we discuss the applicability of powell sabin splines for the numerical solution of partial differential equations defined on domains with polygonal boundary a galerkin type pde discretization is derived for the variable coefficient diffusion equation special emphasis goes to the treatment of dirichlet and neumann boundary conditions finally an error estimator is developed and an adaptive mesh refinement strategy is proposed we illustrate the effectiveness of the approach by means of some numerical experiments

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

continuum aspects of the natural approach abstract the natural finite element approach introduced by argyris in the early sixties is marked essentially by the distinction between rigid body motion and deformation on the one hand and by the description of the latter in compliance with the element purpose and geometry on the other hand for triangular and tetrahedral elements the approach suggests strain and stress measures defined along the sides respectively the edges as homogeneous normal quantities free of shear in the mechanics of continua the respective infinitesimal elements represent minimum configurations to define local deformation in two and three dimensions the paper presents developments of the natural approach on the continuum level within a consistent theoretical framework it is proposed to begin with a reference system of supernumerary coordinates associated with the elementary tetrahedron in the space and with the triangle in the plane vectorial quantities are defined the operations of gradient and divergence are interpreted in this system the natural deformation rate is deduced from the velocity field the stress is introduced as work conjugate measure the condition for local equilibrium is presented in natural quantities as well as the stress definition in association with the resultant forces the set up of material constitutive relations is exemplified for the viscous solid beyond the description of the momentary kinematics as from the velocity field the appearance of finite deformation is considered basing on displacements illustration of the methodology for a plane elastic case is appended

a degree by degree recursive construction of hermite spline interpolants based on the classical hermite spline interpolant h 2 n 1 which is the piecewise interpolation polynomial of class c n 1 and degree 2n 1 a piecewise interpolation polynomial h 2 n of degree 2n is given the formulas for computing h 2 n by h 2 n 1 and computing h 2 n 1 by h 2 n are shown thus a simple recursive method for the construction of the piecewise interpolation polynomial set h j is presented the piecewise interpolation polynomial h 2 n satisfies the same interpolation conditions as the interpolant h 2 n 1 and is an optimal approximation of the interpolant h 2 n 1 some interesting properties are also proved

c 1 hierarchical riesz bases of lagrange type on powell sabin triangulations in this paper we construct c1 continuous piecewise quadratic hierarchical bases on powell sabin triangulations of arbitrary polygonal domains in r2 our bases are of lagrange type instead of the usual hermite type and under some weak regularity assumptions on the underlying triangulations we prove that they form strongly stable riesz bases for the sobolev spaces hs with s 1 5 2 especially the case s 2 is of interest because we can use the corresponding hierarchical basis for preconditioning fourth order elliptic equations leading to uniformly well conditioned stiffness matrices compared to the hierarchical riesz bases by davydov and stevenson hierarchical riesz bases for hs 1 s 5 2 constructive approximation to appear our construction is simpler

propositional dynamic logic of looping and converse is elementarily decidable propositional dynamic logic is a formal system for reasoning about the before after behavior of regular program schemes an extension of propositional dynamic logic which includes both an infinite looping construct and a converse or backtracking construct is considered and it is proved that the satisfiability problem for this logic is elementarily decidable in order to establish this result deterministic two way automata on infinite trees are defined and it is shown how they can be simulated by nondeterministic one way automata the satisfiability problem for propositional dynamic logic of looping and converse is then reduced to the emptiness problem for these two way automata

decidability of finite probabilistic propositional dynamic logics abstract this paper deals with various versions of finite propositional probabilistic dynamic logic we present probabilistic propositional dynamic logic with simple probabilistic estimations and almost regular program language besides the logics previously introduced in the literature we present some natural variations and extensions of these logics we investigate the un decidability of these logics and give a complete picture of decidability and undecidability some of these logics have the finite model property and therefore if they are undecidable they are exactly 10 we show that allowing nesting and probabilistic choice often leads to undecidability

spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem we first develop an efficient spectral galerkin method and a rigorous error analysis for the generalized eigenvalue problems associated to a transmission eigenvalue problem then we present an iterative scheme based on computation of the first transmission eigenvalue to estimate the index of refraction of an inhomogeneous medium we present ample numerical results to demonstrate the effectiveness and accuracy of our approach

c1 quadratic macroelements and c1 orthogonal multiresolution analyses in 2d each triangle of an arbitrary regular triangulation of a polygonal region in r2 is subdivided into twelve subtriangles by using three connecting lines joining three arbitrarily chosen points on its edges three connecting lines from an arbitrarily chosen interior point in the triangle to its three vertices and three connecting lines joining the points on the edges and the interior point in this refinement gd of c1 quadratic finite elements can be constructed in this paper we will give explicit bezier coefficients of elements in terms of the parameters that describe function and first partial derivative values at vertices and values of the normal derivatives at vertices of subtriangles that lie on the edges of consequently the basis and approximation properties of c1 quadratic spline space under refined grid partition gd can be found finally we discuss the construction of c1 orthogonal scaling functions by using c1 quadratic macroelements

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

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 multigrid method for a fourth order diffusion equation with application to image processing in this paper we are concerned with the efficient solution of the elliptic system alpha l nobreak i u nobreak f where l is the biharmonic operator supplemented by higher order boundary conditions this is a core task in image processing arising from the minimization of rotation invariant energy functionals for the purpose of numerical solution we approximate the partial differential equation by finite differences in order to obtain efficient and robust solution methods we discuss how the system can be treated efficiently by multigrid methods finally experimental results are reported that demonstrate the effectiveness of the multigrid approach

a recursive construction of hermite spline interpolants and applications let f k be the hermite spline interpolant of class c k and degree 2k 1 to a real function f which is defined by its values and derivatives up to order k at some knots of an interval a b we present a quite simple recursive method for the construction of f k we show that if at the step k the values of the kth derivative of f are known then f k can be obtained as a sum of f k 1 and of a particular spline g k 1 of class c k 1 and degree 2k 1 beyond the simplicity of the evaluation of g k 1 we prove that it has other interesting properties we also give some applications of this method in numerical approximation

on a hierarchical basis multilevel method with nonconforming p1 elements we develop a hierarchical basis multilevel method for symmetric second order elliptic boundary value problems in two dimensional polygonal domains based on nonconforming p1 triangular elements the main result is that the condition number of the hierarchical discretization is bounded byo k wherek is the number of refinement levels for comparison yserentant s result yields the sharp boundo k 2 for the corresponding hierarchical discretization with conforming linear elements

the additional symmetries for the btl and ctl hierarchies in this paper we construct the additional symmetries for the toda lattice hierarchies of b type and c type the btl and ctl hierarchies and show their algebraic structures are w b w b and w c w c respectively and also we discuss the generating functions of the additional symmetries