boundary trace embedding theorems for variable exponent sobolev spaces we study boundary trace embedding theorems for variable exponent sobolev space w1 p let be an open bounded or unbounded domain in rn satisfying strong local lipschitz condition under the hypotheses that p l 1 infp x supp x n we prove that there is a continuous boundary trace embedding w1 p lq provided q a measurable function on satisfies condition p x q x n 1 p x n p x for x

asymptotic analysis of a semi linear elliptic system in perforated domains well posedness and correctors for the homogenization limit abstract in this study we prove results on the weak solvability and homogenization of a microscopic semi linear elliptic system posed in perforated media the model presented here explores the interplay between stationary diffusion and both surface and volume chemical reactions in porous media our interest lies in deriving homogenization limits upscaling for alike systems and particularly in justifying rigorously the obtained averaged descriptions essentially we prove the well posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrations and then use the structure of the two scale expansions to derive corrector estimates delimitating this way the convergence rate of the asymptotic approximates to the macroscopic limit concentrations our techniques include moser like iteration techniques a variational formulation two scale asymptotic expansions as well as energy like estimates

in any dimension a clamped plate with a uniform weight may change sign abstract positivity preserving properties have been conjectured for the bilaplace dirichlet problem in many versions in this note we show that in any dimension there exist bounded smooth domains such that even the solution of 2 u 1 in with the homogeneous dirichlet boundary conditions u u 0 on is sign changing in two dimensions this corresponds to the kirchhoff love model of a clamped plate with a uniform weight

a three dimensional steady state tumor system abstract the growth of tumors can be modeled as a free boundary problem involving partial differential equations we consider one such model and compute steady state solutions for this model these solutions include radially symmetric solutions where the free boundary is a sphere and nonradially symmetric solutions linear and nonlinear stability for these solutions are determined numerically

detecting non uniqueness of solutions to biharmonic integral equations through svd we consider the singular values of an integral operator and of a corresponding square matrix derived from the integral operator by means of a quadrature formula and a collocation the integral operator and also the matrix depend on a real parameter which may also enter the singular values of the operator and the matrix when a singular value drops to zero for a certain critical value of the parameter the corresponding homogeneous integral equation or matrix equation has a nontrivial solution based on several examples with biharmonic integral operators we conjecture that the order of approximation of the critical value for the matrix is at least equal to the order of the quadrature formula used it is therefore possible with a reasonable accuracy to detect such critical values for the integral operator simply through a singular value decomposition of the matrix derived by a quadrature and collocation

numerical solution for degenerate scale problem arising from multiple rigid lines in plane elasticity abstract this paper studies the degenerate scale problem arising from multiple rigid lines in plane elasticity in the first step the problem should be formulated on a degenerate scale by distribution of body force densities along rigid lines the condition of vanishing displacement along lines is also assumed the coordinate transform with a reduced factor h is performed in the next step the new obtained bie is a particular non homogenous bie defined in the transformed coordinates with normal scale in the normal scale the integral operator is invertible by using two fundamental solutions that are formulated in the normal scale the new obtained bie can be reduced to an equation for finding the factor h finally the degenerate scale is obtained it is proved from computed results that the degenerate scale only depends on the configuration of rigid lines and does not depend on the initial normal scale used in addition the degenerate scale is invariant with respect to the rotation of rigid lines many examples are carried out

boundary element analysis for the helmholtz eigenvalue problems with a multiply connected domain for a helmholtz eigenvalue problem with a multiply connected domain the boundary integral equation approach as well as the boundary element method is shown to yield spurious eigenvalues even if the complex valued kernel is used in such a case it is found that spurious eigenvalues depend on the geometry of the inner boundary demonstrated as an analytical case the spurious eigenvalue for a multiply connected problem with its inner boundary as a circle is studied analytically by using the degenerate kernels and circulants an annular case can be studied analytically in a discrete system and can be treated as a special case the proof for the general boundary instead of the circular boundary is also derived the burton miller method is employed to eliminate spurious eigenvalues in the multiply connected case moreover a modified method considering only the real part formulation is provided five examples are shown to demonstrate that the spurious eigenvalues depend on the shape of the inner boundary good agreement between analytical prediction and numerical results are found

homogenization of degenerate coupled fluid flows and heat transport through porous media we establish a homogenization result for a fully nonlinear degenerate parabolic system with critical growth arising from the heat and moisture flow through a partially saturated porous media existence of a global weak solution of the mesoscale problem is proven by means of a semidiscretization in time a priori estimates and passing to the limit from discrete approximations after that porous material exhibiting periodic spatial oscillations is considered and the two scale convergence as the oscillation period vanishes to a corresponding homogenized problem is rigorously proven

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

numerical examination for degenerate scale problem for ellipse shaped ring region in bie this paper investigates the degenerate scale problem for an ellipse shaped ring region in boundary integral equation bie a homogenous integral equation is introduced the integral equation is reduced to an algebraic equation after discretization the critical value for the degenerate scale can be obtained from the vanishing condition of a determinant it is proved that there are two critical values for the degenerate scale rather than one this finding is first proposed in the paper two particular problems with known solutions are examined numerically the loadings applied on the exterior boundary may result in a resultant force in the x direction or in the y direction the improper numerical solutions have been found once the real size approaches the critical value two techniques for avoiding the improper solutions are suggested the techniques depend on the appropriate choice of the used size or adding a constant in a kernel of the integral equation it is proved that both techniques will give accurate numerical results numerical examinations for the problem are emphasized in the paper copyright 2007 john wiley sons ltd

analytical study and numerical experiments for degenerate scale problems in the boundary element method for two dimensional elasticity for a plane elasticity problem the boundary integral equation approach has been shown to yield a non unique solution when geometry size is equal to a degenerate scale in this paper the degenerate scale problem in the boundary element method bem is analytically studied using the method of stress function for the elliptic domain problem the numerical difficulty of the degenerate scale can be solved by using the hypersingular formulation instead of using the singular formulation in the dual bem a simple example is shown to demonstrate the failure using the singular integral equations of dual bem it is found that the degenerate scale also depends on the poisson s ratio by employing the hypersingular formulation in the dual bem no degenerate scale occurs since a zero eigenvalue is not imbedded in the influence matrix for any case

the degenerate scale problem for the laplace equation and plane elasticity in a multiply connected region with an outer circular boundary this paper investigates the degenerate scale problem for the laplace equation and plane elasticity in a multiply connected region with an outer circular boundary inside the boundary there are many voids with arbitrary configurations the problem is analyzed with a relevant homogenous bie boundary integral equation it is assumed that all the inner void boundary tractions are equal to zero and tractions on the outer circular boundary are constant therefore all the integrations in bie are performed on the outer circular boundary only by using the relation z conjg z a a or conjg z a a z on the circular boundary with radius a all integrals can be reduced to an integral for complex variable and they can be integrated in closed form the degenerate scale a 1 is found in the laplace equation and in plane elasticity regardless of the void configuration

analytical study and numerical experiments for radiation and scattering problems using the chief method in this paper the well known failure in numerical computations of the exterior helmholtz integral equation at certain characteristic frequencies is investigated analytically and numerically the combined helmholtz integral equation formulation chief method is a very popular technique that can overcome the non unique problem but this method breaks down if the interior points are not properly chosen the chief method in conjunction with the singular value decomposition svd technique is an easy and efficient method to ensure a unique solution for the exterior problem based on the circulant properties and degenerate kernels an analytical scheme in discrete system of a cylinder is achieved the optimum numbers and proper positions for the collocation points in the interior domain are analytically studied and suggested in the numerical scheme numerical experiments are designed to verify the analytical results one analytical example for a circular cylinder and one numerical example for a square rod are demonstrated to show the validity of the chief method

asymptotic analysis of perforated plates and membranes part 2 static and dynamic problems for large holes abstract static and dynamic problems for the elastic plates and membranes periodically perforated by holes of different shapes are solved using the combination of the singular perturbation technique and the multi scale asymptotic homogenization method the problems of bending and vibration of perforated plates are considered using the asymptotic homogenization method the original boundary value problems are reduced to the combination of two types of problems first one is a recurrent system of unit cell problems with the conditions of periodic continuation and the second problem is a homogenized boundary value problem for the entire domain characterized by the constant effective coefficients obtained from the solution of the unit cell problems in the present paper the perforated plates with large holes are considered and the singular perturbation method is used to solve the pertinent unit cell problems matching of limiting solutions for small and large holes using the two point pade approximants is also accomplished and the analytical expressions for the effective stiffnesses of perforated plates with holes of arbitrary sizes are obtained

an asymptotic property of degenerate scales for multiple holes in plane elasticity the solution of an exterior dirichlet boundary value problem of plane isotropic elasticity by the boundary integral equation of the first kind obtained from the somigliana identity is considered the logarithmic function appearing in the integral kernel may cause that the operator is non invertible such a situation occurs if the size of the boundary coincides with a degenerate scale for a certain form of the fundamental solution used a technique for the evaluation of the degenerate scales for an asymptotic case of an exterior domain with holes placed far one from another is discussed and analyzed the examples provide results of particular cases and assess suitability of the proposed technique in relation to numerical calculation of degenerate scales by the boundary element method

asymptotic error expansions and splitting extrapolation algorithm for two classes of two dimensional cauchy principal value integrals abstract this paper proposes numerical quadrature rules for two dimensional cauchy principal value integrals of the forms f x y x s 2 y t 2 d y d x and f x y x s y t d y d x the derivation of these quadrature rules is based on the euler maclaurin error expansion of a modified trapezoidal rule for one dimensional cauchy singular integrals the corresponding error estimations are investigated and the convergence rates o h m 2 h n 2 are obtained for the proposed quadrature rules where hm and hn are partition sizes in x and y directions is a positive integer determined by integrand to further improve accuracy a splitting extrapolation algorithm is developed based on the asymptotic error expansions several numerical tests are performed to verify the effectiveness of the proposed methods

solution for dissimilar elastic inclusions in a finite plate using boundary integral equation method abstract this paper studies the boundary value problem for a finite plate containing two dissimilar inclusions the matrix and the two inclusions have different elastic properties the loadings applied along the outer boundary are in equilibrium the mentioned problem is decomposed into three boundary value problems bvps two of them are interior bvp for the elastic inclusions while the other is a bvp for the triply connected region three problems are connected together through the common displacements and tractions along the interface boundaries explicit form for the complex variable boundary integral equation cvbie is derived after discretization of relevant bies the solutions are evaluated numerically three numerical examples for different elastic constant combinations are provided

analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real part dual bem it has been found recently that the multiple reciprocity method mrm chen and wong engng anal boundary elements 1997 20 1 25 33 chen processings of the fourth world congress on computational mechanics onate e idelsohn sr eds argentina 1998 106 chen and wong j sound vibration 1998 217 1 75 95 or real part bem liou chen and chen j chinese inst civil hydraulics 1999 11 2 299 310 in chinese results in spurious eigenvalues for eigenproblems if only the singular ut or hypersingular lm integral equation is used in this paper a circular cavity is considered as a demonstrative example for an analytical study based on the framework of the real part dual bem the true and spurious eigenvalues can be separated by using singular value decomposition svd to understand why spurious eigenvalues occur analytical derivation by discretizing the circular boundary into a finite degree of freedom system is employed resulting in circulants for influence matrices based on the properties of the circulants we find that the singular integral equation of the real part bem for a circular domain results in spurious eigenvalues which are the zeros of the bessel functions of the second kind y k while the hypersingular integral equation of the real part bem results in spurious eigenvalues which are the zeros of the derivative of the bessel functions of the second kind yn k it is found that spurious eigenvalues exist in the real part bem and that they depend on the integral representation one uses singular or hypersingular single layer or double layer no matter what the given types of boundary conditions for the interior problem are furthermore spurious modes are proved to be trivial in the circular cavity through analytical derivations numerically they appear to have the same nodal lines of the true modes after normalization with respect to a very small nonzero value two examples with a circular domain including the neumann and dirichlet problems are presented the numerical results for true and spurious eigensolutions match very well with the theoretical prediction copyright 2000 john wiley sons ltd

hypersingular integral equation for multiple curved cracks problem in plane elasticity the complex variable function method is used to formulate the multiple curved crack problems into hypersingular integral equations these hypersingular integral equations are solved numerically for the unknown function which are later used to find the stress intensity factor sif for the problem considered numerical examples for double circular arc cracks are presented

exact degenerate scales in plane elasticity using complex variable methods a recent work has shown that using conformal mapping can lead to exact values of the degenerate scales in plane elasticity we elaborate on this work by introducing some algebraic tools when this conformal mapping is a rational fraction transforming the outside of the unit circle into the outside of the considered domain using these tools new cases are solved including shortened hypotrochoid arc of circle new approximates of equilateral triangle and square or symmetric joukowski profiles another method makes it possible to obtain the degenerate scales for plane elasticity from the degenerate scale for laplace s equation for some multiply connected sets the cases of segments on a line or of arcs of circle with a n fold symmetry in these last cases the exact values of the degenerate scales are obtained when the degenerate scale for the laplace problem is known

derivation and analytical investigation of three direct boundary integral equations for the fundamental biharmonic problem we derive and investigate three families of direct boundary integral equations for the solution of the plane fundamental biharmonic boundary value problem these three families are fairly general so that they as special cases encompass various known and applied equations as demonstrated by giving many references to the literature we investigate the families by analytical means for a circular boundary curve where the radius is a parameter we find for all three combinations of equations i that the solution of the equations is non unique for one or more critical radius radii and ii that this lack of uniqueness can always be removed by combining the integral equations with a suitable combination of one or more supplementary condition s we conjecture how the results obtained can or cannot be generalized to other boundary curves through the concept logarithmic capacity a few published general results about uniqueness are compared with our findings

second kind integral equation formulation for the modified biharmonic equation and its applications a system of fredholm second kind integral equations skies is constructed for the modified biharmonic equation in two dimensions with gradient boundary conditions such boundary value problem arises naturally when the incompressible navier stokes equations are solved via a semi implicit discretization scheme and the resulting boundary value problem at each time step is then solved using the pure stream function formulation the advantages of such an approach greengard and kropinski 1998 14 are two fold first the velocity is automatically divergence free and second complicated nonlocal boundary conditions for the vorticity are avoided our construction though similar to that of farkas 1989 10 for the biharmonic equation is modified such that the skie formulation has low condition numbers for large values of the parameter we illustrate the performance of our numerical scheme with several numerical examples finally the scheme can be easily coupled with standard fast algorithms such as fft fast multipole methods greengard and rokhlin 1987 15 or fast direct solvers ho and greengard 2012 martinsson and rokhlin 2005 17 25 to achieve optimal complexity bringing accurate large scale long time fluid simulations within practical reach

the trefftz method using fundamental solutions for biharmonic equations in this paper the trefftz method of fundamental solution fs called the method of fundamental solution mfs is used for biharmonic equations the bounds of errors are derived for the mfs with almansi s fundamental solutions denoted as the mafs in bounded simply connected domains the exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions respectively the stability analysis of the mafs is also made for circular domains numerical experiments are carried out for both smooth and singularity problems the numerical results coincide with the theoretical analysis made when the particular solutions satisfying the biharmonic equation can be found the method of particular solutions mps is always superior to the mfs and the mafs based on numerical examples however if such singular particular solutions near the singular points do not exist the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points based on the computed results the mfs using the greedy adaptive techniques may provide more accurate solutions for singularity problems moreover the numerical solutions by the mafs with almansi s fs are slightly better in accuracy and stability than those by the traditional mfs hence the mafs with the afs is recommended for biharmonic equations due to its simplicity

analytical derivation and numerical experiments of degenerate scales for regular n gon domains in two dimensional laplace problems degenerate scale of a regular n gon domain is studied by using the boundary element method bem and complex variables degenerate scale stems from either the non uniqueness of bie using the logarithmic kernel or the conformal radius of unit logarithmic capacity in the complex variables analytical formula and numerical results for the degenerate scale are obtained by using the conformal radius and boundary element program respectively analytical formula of the degenerate scale contains the gamma function for the gamma contour which can be derived from the schwarz christoffel mapping based on the dual bem the rank deficiency mathematical mode due to the degenerate scale mathematics is imbedded in the left unitary vector for the influence matrices of weakly singular u kernel and strongly singular t kernel integral operators on the other hand we obtain the common right unitary vector corresponding to a rigid body mode physics in the influence matrices of strongly singular t kernel and hypersingular m kernel operators after using the singular value decomposition to deal with the problem of non unique solution the constraint of boundary flux equilibrium instead of rigid body term cheef and hypersingular bie is added to promote the rank of influence matrices to be full rank null field for the exterior domain and interior nonzero field are analytically derived and numerically verified for the normal scale while the interior null field and nonzero exterior field are obtained for the homogeneous dirichlet problem in the case of the degenerate scale it is found that the contour of nonzero exterior field for the degenerate scale using the bem matches well with that of schwarz christoffel transformation both analytical and numerical results agree well in the demonstrative examples of right triangle square regular 5 gon and regular 6 gon it is straightforward to extend to general regular n gon case

vibrations of thin plates with small clamped patches eigenvalues of fourth order elliptic operators feature prominently in stability analysis of elastic structures this paper considers out of plane modes of vibration of a thin elastic plate perforat

asymptotic analysis of perforated plates and membranes part 1 static problems for small holes abstract static problems for the elastic plates and rods periodically perforated by small holes of different shapes are solved using the asymptotic approach based on the combination of the asymptotic technique and the multi scale homogenization method using the asymptotic homogenization method the original boundary value problem is reduced to the combination of two types of problems first one is a recurrent system of unit cell problems with the conditions of periodic continuation and the second problem is a homogenized boundary value problem for the entire domain characterized by the constant effective coefficients obtained from the solution of the unit cell problems the combination of the perturbation method and the technique of successive approximations is applied for the solution of the unit cell problems taking into the account small size of holes the method of perturbation of the shape of the boundary and the schwarz alternating method are used the problems of torsion of a rod with perforated cross section deflection of the perforated membrane loaded by a normal load and bending of perforated plates with circular and square holes are considered consecutively the error of the applied asymptotic techniques is estimated and the high accuracy of the obtained solutions is demonstrated

approximation of dirichlet eigenvalues on domains with small holes abstract approximation formulas for the eigenvalues of the laplacian with dirichlet boundary conditions on domains with small holes are derived and discussed the corresponding results for the free and the supported vibrating plate follow a conceptually simple proof is given based on courant s min max principle the author s approximation for the capacity of small balls leads to a highly accurate eigenvalue approximation formula for domains with spherical holes general holes are treated by means of a harmonic correction method an isoperimetric inequality relating capacity to volume and a poincare inequality for capacity potentials in addition we provide l bounds for the eigenfunctions

the transition to a point constraint in a mixed biharmonic eigenvalue problem the mixed order eigenvalue problem delta delta 2 u delta u lambda u 0 with delta 0 modeling small amplitude vibrations of a thin plate is analyzed in a bounded two dimensional domain omega that contains a single small hole of radius varepsilon centered at some x 0 in omega clamped conditions are imposed on the boundary of omega and on the boundary of the small hole in the limit varepsilon to 0 and for delta mathcal o 1 the limiting problem for u must satisfy the additional point constraint u x 0 0 to determine how the eigenvalues of the laplacian in a domain with a small hole are perturbed by adding the small fourth order term delta delta 2 u together with an additional boundary condition on partial omega and on the hole boundary the asymptotic behavior of the eigenvalues of the mixed order eigenvalue problem are studied in the dual limit varepsilon to 0 and delta to 0 leading order behaviors of eigenvalues are determined for three ranges

an svd analysis of linear algebraic equations derived from first kind integral equations abstract by means of singular value decomposition svd we investigate the systems of linear algebraic equations which are derived for numerical solution of first kind fredholm integral equations arising in two dimensional potential theory in order to derive a numerical model which has the same features and peculiarities as the underlying integral equation we apply the galerkin method with orthonormal basis functions the linear equations are studied with respect to 1 conditioning 2 accuracy of the computed solution 3 effective rank and 4 comparison of nullspaces as a practical example we then use our numerical method to investigate the equations of a specific geometry for which the analytical solution is not known

computing steady state solutions for a free boundary problem modeling tumor growth by stokes equation we consider a free boundary problem modeling tumor growth where the model equations include a diffusion equation for the nutrient concentration and the stokes equation for the proliferation of tumor cells for any positive radius r it is known that there exists a unique radially symmetric stationary solution the proliferation rate m and the cell to cell adhesiveness c are two parameters for characterizing aggressiveness of the tumor we compute symmetry breaking bifurcation branches of solutions by studying a polynomial discretization of the system by tracking the discretized system we numerically verified a sequence of m c symmetry breaking bifurcation branches furthermore we study the stability of both radially symmetric and radially asymmetric stationary solutions

degenerate scales for boundary value problems in anisotropic elasticity abstract degenerate scales usually refer to a size effect which causes non unique solutions of boundary integral equations for certain type of boundary value problems with a unique solution they are closely connected to the presence of a logarithmic function in the integral kernel of the single layer potential operator the equations of the elasticity theory provide one of the known application fields where degenerate scales appear the paper discusses conditions and formula for controlling and detection of the degenerate scales in the case of fully anisotropic analysis no restrictions are considered for the material only the loading should cause two dimensional deformation of the anisotropic body a technique for the evaluation of the degenerate scales is discussed and tested the examples provide results of special simple cases and demonstrate suitability of the proposed technique in relation to calculation of degenerate scales by numerical solution of pertinent boundary integral equation by the boundary element method

degenerate scale problem when solving laplace s equation by bem and its treatment in this paper laplace problems are solved by using the dual boundary element method bem it is found that a degenerate scale problem occurs if the conventional bem is used in this case the influence matrix is rank deficient and numerical results become unstable both the circular and elliptical bars are studied analytically in the continuous system in the discrete system the fredholm alternative theorem in conjunction with the svd singular value decomposition updating documents is employed to sort out the spurious mode which causes the numerical instability three regularization techniques method of adding a rigid body mode hypersingular formulation and cheef combined helmholtz exterior integral equation formulation concept are employed to deal with the rank deficiency problem the addition of a rigid body term c in the fundamental solution is proved to shift the original degenerate scale to a new degenerate scale by a factor e c the torsion rigidities are obtained and compared with analytical solutions numerical examples including elliptical square and triangular bars were demonstrated to show the failure of conventional bem in case of the degenerate scale after employing the three regularization techniques the accuracy of the proposed approaches is achieved copyright 2004 john wiley sons ltd

revisit of a degenerate scale boundary element method bem has been employed in engineering analysis since 1956 it has been widely applied in the engineering however the bem biem may result in an ill conditioned system in some special situations such as the degenerate scale the degenerate scale also relates to the logarithmic capacity in the modern potential theory in this paper three indexes to detect the degenerate scale and five regularization techniques to circumvent the degenerate scale are reviewed and a new self regularization technique by using the bordered matrix is proposed both the analytical study and the bem implementation are addressed for the analytical study we employ the riemann conformal mapping of complex variables to derive the unit logarithmic capacity the degenerate scale can be analytically derived by using the conformal mapping as well as numerical detection by using the bem in the theoretical aspect we prove that unit logarithmic capacity in the riemann conformal mapping results in a degenerate scale we revisit the fredholm alternative theorem by using the singular value decomposition svd the discrete system and explain why the direct bem and the indirect bem are not indeed equivalent in the solution space besides a zero index by using the free constant in fichera s approach is also proposed to examine the degenerate scale according to the relation between the svd structure and fichera s technique we numerically provide a new self regularization method in the matrix level finally a semi circular case and a special shape case are designed to demonstrate the validity of six regularization techniques

a new method for true and spurious eigensolutions of arbitrary cavities using the combined helmholtz exterior integral equation formulation method integral equation methods have been widely used to solve interior eigenproblems and exterior acoustic problems radiation and scattering it was recently found that the real part boundary element method bem for the interior problem results in spurious eigensolutions if the singular ut or the hypersingular lm equation is used alone the real part bem results in spurious solutions for interior problems in a similar way that the singular integral equation ut method results in fictitious solutions for the exterior problem to solve this problem a combined helmholtz exterior integral equation formulation method cheef is proposed based on the cheef method the spurious solutions can be filtered out if additional constraints from the exterior points are chosen carefully finally two examples for the eigensolutions of circular and rectangular cavities are considered the optimum numbers and proper positions for selecting the points in the exterior domain are analytically studied also numerical experiments were designed to verify the analytical results it is worth pointing out that the nodal line of radiation mode of a circle can be rotated due to symmetry while the nodal line of the rectangular is on a fixed position

approximate solution for degenerate scale problem of two rigid lines in series in plane elasticity abstract by using the technique of coordinate transform the degenerate scale problem of two rigid lines in series can be solved in the normal scale it is assumed that the distance between two rigid lines is much longer than the size of the rigid lines in this case the formulated equation can be significantly simplified and an approximate solution is obtainable numerical examination proves that the approximate solution can also be used to a wider range with a close distance between two lines

numerical study of three multilevel preconditioners for solving 2d unsteady navier stokes equations abstract numerical experiments for solving 2d navier stokes equations in the stream function formulation described by a fourth order equation for which is a natural formulation from a physical point of view are presented we use standard finite difference discretization for fourth order elliptic equations where the non linear convective terms are taken from the preceding time step the resulting symmetric linear systems of equations are very ill conditioned we overcome this difficulty by a proper preconditioning technique we investigate the performance of three preconditioners constructed on the basis of optimal order multilevel preconditioners for second order elliptic operators the numerical tests presented illustrate the limitations and advantages of the proposed preconditioners when we vary the reynolds number the mesh size in space and the time step

algebraic multilevel preconditioning methods i a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance

hybrid v cycle algebraic multilevel preconditioners we consider an algebraic derivation of multilevel preconditioners which are based on a sequence of finite element stiffness matrices they cor respond to a sequence of triangulations obtained by successive refinement and the associated finite element discretizations of second order selfadjoint elliptic boundary value problems the stiffness matrix at a given discretization level is partitioned into a natural hierarchical two level two by two block form then it is factored into block triangular factors the resulting schur complement is then replaced approximated by the stiffness matrix on the preceding coarser level this process is repeated successively for a fixed number ko 1 of steps after each cq steps the preconditioner so derived is corrected by a certain polynomial approximation a properly scaled and shifted chebyshev matrix polynomial which involves the preconditioner and the stiffness matrix at the considered level the hybrid f cycle preconditioner thus derived is shown to be of optimal order of complexity for 2 d and 3 d problem domains the rela tive condition number of the preconditioner is bounded uniformly with respect to the number of levels and with respect to possible jumps of the coefficients of the considered elliptic bilinear form as long as they occur only across edges faces in 3 d of elements from the coarsest triangulation in addition an adap tive implementation of our hybrid k cycle preconditioners is proposed and its practical behavior is demonstrated on a number of test problems

performance of certain iterative methods in solving implicit difference schemes for 2 d navier stokes equations the performance of three iterative methods local sor preconditioned generalized conjugate gradient methods such as orthomin 1 and the gcg ls method with preconditioning matrices derived by incomplete modified and unmodified pointwise factorizations of the matrix corresponding to implicit finite difference schemes for the 2 d navier stokes equations is compared the relative merits of these methods in terms of cpu time average number of iterations with respect to the reynolds number and discretization parameters are emphasized

use of simple solutions for boundary integral methods in elasticity and fracture analysis the simple solutions or indirect method of analysing cauchy and hypersingular integrands in the gradient flux or traction boundary integral equation bie is applied to linear elastic fracture analysis because of the geometric singularity of the crack surface application of the simple solutions formulas on the crack face requires integration over a temporary closure surface rather than the remainder of the body closure surface constructions are exhibited for crack surfaces allowing the gradient bie to be applied as a constraint equation on a crack surface where the primary bie is degenerate computational results are given for two benchmark fracture problems

semilinear elliptic problems with mixed dirichlet neumann boundary conditions this work deals with the analysis of problems u uq urin u 0in b u 0on where b u means mixed boundary conditions we prove results about existence multiplicity and a priori estimates

boundary integral equation method for periodic dissimilar elastic inclusions in an infinite plate abstract this paper provides a theoretical derivation and a numerical solution for an infinite plate with periodic dissimilar inclusions the matrix and periodic inclusions have different elastic properties and a remote loading is applied for the infinite matrix the original problem is decomposed into two problems one is the interior boundary value problem bvp and the other is the exterior bvp both bvps are formulated in the form of complex variable boundary integral equation cvbie in addition both bies are solved numerically after discretization of the integral equations for the exterior bvp with periodic notches the remainder estimation technique ret is suggested to evaluate the influence to the central notch from many from n th to infinity neighboring notches this will significantly improve the efficiency of solution two numerical examples are provided in the examples the ratio of the two shear moduli of elasticity changes from near 0 1 d 6 0 1 0 5 1 2 to 10

properties of the principal eigenvalues of a general class of non classical mixed boundary value problems in this paper we characterize the existence of principal eigenvalues for a general class of linear weighted second order elliptic boundary value problems subject to a very general class of mixed boundary conditions our theory is a substantial extension of the classical theory by p hess and t kato 1980 comm partial differential equations5 999 1030 in obtaining our main results we must give a number of new results on the continuous dependence of the principal eigenvalue of a second order linear elliptic boundary value problem with respect to the underlying domain and the boundary condition itself these auxiliary results complement and in some sense complete the theory of d daners and e n dancer 1997 j differential equations138 86 132 the main technical tool used throughout this paper is a very recent characterization of the strong maximum principle in terms of the existence of a positive strict supersolution due to h amann and j lopez gomez 1998 j differential equations146 336 374

block implicit multigrid solution of navier stokes equations in primitive variables abstract a calculation procedure for rapid computation of steady multidimensional viscous flows is presented the method solves the navier stokes equations in primitive variables using a coupled block implicit multigrid procedure the procedure is applicable to finite difference formulations using staggered locations of the flow variables a smoothing technique called symmetrical coupled gauss seidel scgs is proposed and is empirically observed to provide good smoothing rates the viscous flow in a square cavity with a moving top wall is calculated for a range of reynolds numbers calculations with finite difference grids as large as 321 321 nodes have been made to test the accuracy and efficiency of the calculation scheme the cpu times for these calculations are observed to be significantly smaller than other solution algorithms with primitive variable formulation the calculated flow fields in the cavity are in good agreement with earlier studies of the same flow situation