immersed interface finite element methods for elliptic interface problems with nonhomogeneous jump conditions in this work a class of new finite element methods called immersed interface finite element methods is developed to solve elliptic interface problems with nonhomogeneous jump conditions simple non body fitted meshes are used a single function that satisfies the same nonhomogeneous jump conditions is constructed using a level set representation of the interface with such a function the discontinuities across the interface in the solution and flux are removed and an equivalent elliptic interface problem with homogeneous jump conditions is formulated special finite element basis functions are constructed for nodal points near the interface to satisfy the homogeneous jump conditions error analysis and numerical tests are presented to demonstrate that such methods have an optimal convergence rate these methods are designed as an efficient component of the finite element level set methodology for fast simulation of interface dynamics that does not require remeshing

uniform asymptotic stability in impulsive perturbed systems of differential equations abstract in the present paper the uniform asymptotic stability of the zero solution of perturbed impulsive systems of differential equations with unfixed moments of impulse action is investigated the direct method of lyapunov is used but the auxiliary functions possess first kind discontinuities on the impulsive hypersurfaces

optimal error bounds for partially penalized immersed finite element methods for parabolic interface problems abstract in lin et al 2015 a group of typical partially penalized immersed finite element ppife methods were developed for solving interface problems of the parabolic equations and its authors proved that these ppife methods had the optimal o h convergence rate in an energy norm under a sub optimal piecewise h 3 regularity assumption in this article we report our reanalysis of these ppife methods our proofs are based on the optimal error bounds recently derived in guo et al 2019 for elliptic interface problems and we are able to show that these ppife methods have the optimal convergence not only in an energy norm but also in the usual l 2 norm with the assumption that the exact solution possesses the standard piecewise h 2 regularity instead of the excessive piecewise h 3 regularity in the space variable numerical results are also presented to validate the reported error analysis

immersed finite element methods for 4th order differential equations we propose three new finite element methods for solving boundary value problems of 4th order differential equations with discontinuous coefficients typical differential equations modeling the small transverse displacement of a beam and a thin plate formed by multiple uniform materials are considered one important feature of these finite element methods is that their meshes can be independent of the interface between different materials finite element spaces based on both the conforming and mixed formulations are presented numerical examples are given to illustrate capabilities of these methods

the partition of unity method for the elastically supported beam abstract the partition of unity method pum is used to solve the timoshenko beam with elastic support some important features of this new method are addressed but the main concern is to overcome locking and to resolve boundary layer we prove that by a proper selection of the local basis functions the method is free of locking at the thin beam limit and exhibits no numerical boundary layer for strong elastic support optimal convergent rate is established in the energy norm and it is uniformly valid with respect to the thickness of the beam and toughness of the elastic support furthermore the computed shear stress is also convergent uniformly with optimal rate

a trilinear immersed finite element method for solving the electroencephalography forward problem this article describes a trilinear immersed finite element method for solving the electroencephalography forward problem which is a three dimensional elliptic interface problem in the head geometry the method uses hexahedral cartesian meshes independent of the interfaces between head tissues thus avoiding the sometimes difficult task of generating geometry fitting meshes the interfaces are represented with level sets and the finite element space is locally modified to better approximate the discontinuities of the solution numerical results show that this method achieves the same accuracy as the standard linear finite element method with geometry fitting mesh

random initial imperfections of structures abstract this paper is a theoretical study on random initial imperfections of structures the explicit form of probability density function of the load bearing capacity critical load of structures is derived for random initial imperfections based on a decomposition of the space of imperfection vectors into two orthogonal subspaces the subspace that asymptotically affects the load bearing capacity and the other that does not tight bounds on the range of load bearing capacity are presented for various types of simple critical points by means of the asymptotic theory of statistics we show the inefficiency of a conventional random method that approximates the minimum loadhearing capacity by the minimum load for a number of random initial imperfections the theoretical and empirical probability distribution functions for simple truss structures are compared to show the validity and effectiveness of the present method

perturbing uniformly stable nonlinear scalar delay differential equations on considere l equation differentielle a retard perturbee x t f t x t g t x t ou g o s r s x r x on donne des conditions sur g qui assurent que la solution zero est uniformement stable sous certaines conditions

probabilistic eigenvalue buckling analysis solved through the ratio of polynomial response surface an efficient procedure for the reliability analysis of frame structures with respect to the buckling limit state is proposed under the assumption that no imperfections are present and that the elastic parameters are uncertain and modeled as random variables the approach allows a deeper investigation of structures which are not sensitive to imperfections the procedure relies on a response surface method adopting simple ratio of polynomials without cross terms as performance function such a relationship approximates analytically the dependence between the buckling load and the basic variables furnishing a limit state equation which is very close to the exact one when a proper experimental design is adopted in this way a monte carlo simulation applied to the response surface leads to a good approximation with low computational effort several numerical examples show the accuracy and effectiveness of the method varying structural complexity correlation between basic variables and their distribution

uniform asymptotic stability for perturbed neutral delay differential equations abstract one dimensional perturbed neutral delay differential equations of the form x t p t x t f t x t g t x t are considered assuming that f satisfies v t m f t v t m where m max 0 max s r 0 s a typical result is the following if g t w t and t t 1 w s ds 0 then the zero solution is uniformly asymptotically stable providing that the zero solution of the corresponding equation without perturbation x t p t x t f t x t is uniformly asymptotically stable some known results associated with this equation are extended and improved

uniform asymptotic stability in perturbed differential equations on etudie la stabilite asymptotique uniforme de la solution zero des equations differentielles perturbees x f t x g t x et si t f t x t g t x t

statistics of normally distributed initial imperfections abstract this paper offers a theoretical study on the probabilistic nature of critical loads buckling loads of structures subject to normally distributed initial imperfections explicit form of probability density function of critical loads are derived for various types of critical points double bifurcation points of structures with regular polygonal symmetry are dealt with by means of the group theoretic bifurcation theory the distribution of minimum values of the critical loads is investigated to present a statistical design index the theoretical and empirical probability density functions for simple structures are compared to show the validity and effectiveness of this method the method is quite efficient when it is directly applicable otherwise the explicit forms at least can greatly supplement the inefficiency of the conventional random method

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

a stable generalized finite element method sgfem of degree two for interface problems abstract generalized or extended finite element methods gfem xfem of degree 1 for interface problems have been reported in the literature they i yield optimal order of convergence i e o h h is a discretization parameter ii are stable in a sense that conditioning is not worse than that of the standard fem and iii are robust in that the conditioning does not deteriorate as interface curves are close to boundaries of underlying elements however higher order gfem xfem with the properties i iii have not been successfully addressed yet various enrichment schemes for gfem xfem based on d or d p k d is a distance function or the absolute value of level set function and p k is the polynomial basis of degree k have been reported recently to obtain higher order convergence but they are not stable or robust in general in fact they even may not yield the optimal orders of convergence in this paper we propose a stable gfem xfem of degree 2 sgfem2 for the interface problems where we used the enrichment scheme based on d 1 x y instead of d or d 1 x y x 2 x y y 2 in the literature we prove that the sgfem2 yields the optimal order of convergence i e o h 2 for the interface problems with curved smooth interfaces a local principal component analysis technique has been proposed which ensures that the sgfem2 is stable and robust numerical experiments for straight and curved interfaces have been presented to illuminate these properties

a conforming enriched finite element method for stokes interface problems abstract a new conforming enriched finite element method is presented for stokes interface problems with interface unfitted meshes the conforming enriched finite element pair is constructed based on the mini element pair a ghost penalty term is used in the standard discretization form as a stabilization term an inf sup stability result is derived which is uniform with respect to the mesh size our method does not limit the viscosity coefficient to be a piecewise constant finite element errors are proved to be optimal numerical experiments are carried out to validate theoretical results

modeling and an immersed finite element method for an interface wave equation abstract the electromagnetic field which is governed by maxwell s equation plays a key role in plasma simulation in this article we first derive the interface conditions when we rewrite the interface maxwell s equation whose problem domain involves complex media such as objects of different materials into a parabolic hyperbolic type of interface model which is a modified wave equation by adding a first order time derivative term due to the lossy medium based on the interface conditions and the existing bilinear immersed finite element space for the interface poisson equation we propose an immersed finite element method for the spatial discretization of this parabolic hyperbolic interface equation on a cartesian mesh independent of the interface then we use a second order finite difference method for the temporal discretization in order to develop the full discretization scheme compared with the unstructured body fitting mesh which is needed by the traditional finite element method for interface problems the cartesian mesh independent of the interface will significantly benefit the plasma simulation in the electromagnetic field especially the particle models such as the particle in cell method this method provides a new efficient way to study varying electromagnetic field with objects of different materials on a cartesian mesh independent of the interface hence builds a solid foundation for the further study on the motion of plasma in this electromagnetic field numerical examples are provided to demonstrate the features of the proposed method

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

interpolating stabilized moving least squares mls approximation for 2d elliptic interface problems abstract the main aim of the current paper is to propose a new truly meshless numerical technique to solve the one and two dimensional elliptic interface problems the employed meshless approach is based on a new class of mls approximation the numerical procedure is based on the interpolating stabilized mls ismls approximation the new shape functions that have been made by the ismls technique have the kronecker property thus the dirichlet boundary conditions can be applied directly in the current investigation we propose a meshless collocation method for solving elliptic interface problems based on the shape functions of ismls technique several test problems with sufficient complexity have been studied to check the accuracy and efficiency of the new numerical procedure moreover test problems show the acceptable accuracy and efficiency of the proposed scheme

local discontinuous galerkin methods for parabolic interface problems with homogeneous and non homogeneous jump conditions abstract in this paper we present the local discontinuous galerkin ldg methods for solving second order parabolic interface problems in two dimensional convex polygonal domains with homogeneous and non homogeneous jump conditions respectively we start analyzing from the homogeneous problems after giving the semi discrete ldg method we derive its stability and a prior error estimate then after a series of derivation we obtain the ldg method for the non homogeneous problems which takes the same formulation as the homogeneous case with a special choice of numerical fluxes that incorporate the jump conditions across interface so its analysis can be easily obtained within the framework developed for the homogeneous case it is an advantage of the ldg method that it provides a natural framework to enforce the discontinuities in both the potential and the flux weakly in the discrete formulation provided that the triangulation of the domain is fitted to the interface numerical experiments are conducted to confirm our theoretical analysis with the help of the explicit temporal discretization and show that the ldg methods have the optimal and suboptimal convergence rate in the l 2 norms for the potential and the flux respectively in addition they also indicate that the ldg methods possess the same convergence properties in the l norms what is more to avoid the strict time step restriction of explicit schemes the implicit integration factor iif method is employed which not only relaxes the time step to t o h m i n but also remains the important property of ldg method that the computation can proceed element by element and avoid solving a global system of algebraic equations as the standard implicit schemes do we provide numerical examples to illustrate that the numerical scheme combined ldg method with iif method indeed reduces the computational cost greatly

immersed finite element methods for unbounded interface problems with periodic structures interface problems arise in many physical and engineering simulations involving multiple materials periodic structures often appear in simulations with large or even unbounded domain such as magnetostatic electrostatic field simulations immersed finite element ife methods are efficient tools to solve interface problems on a cartesian mesh which is desirable to many applications like particle in cell simulation of plasma physics in this article we develop an ife method for an interface problem with periodic structure on an infinite domain to cope with the periodic boundary condition we modify the stiffness matrix of the ife method the new matrix is maintained symmetric positive definite so that the linear system can be solved efficiently numerical examples are provided to demonstrate features of this method

partially penalized immersed finite element methods for elliptic interface problems this article presents new immersed finite element ife methods for solving the popular second order elliptic interface problems on structured cartesian meshes even if the involved interfaces have nontrivial geometries these ife methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in ife functions with the enhanced stability due to the added penalty not only can these ife methods be proven to have the optimal convergence rate in an energy norm provided that the exact solution has sufficient regularity but also numerical results indicate that their convergence rates in both the h 1 norm and the l 2 norm do not deteriorate when the mesh becomes finer which is a shortcoming of the classic ife methods in some situations trace inequalities are established for both linear and bilinear ife functions that are not only critical for the error analysis of these new ife methods but are also of a great potential to be useful in error analysis for other

a posteriori error analysis of the crank nicolson finite element method for linear parabolic interface problems a reconstruction approach abstract this article studies a residual based a posteriori error analysis for the crank nicolson time stepping finite element method for a linear parabolic interface problem in a bounded convex polygonal domain in r 2 a piecewise linear finite element space is used in space that is allowed to change in time and a modified crank nicolson approximation is applied for the time discretizations we employ a space time reconstruction that is piecewise quadratic in time and the clement type interpolation estimates to derive optimal order in time and an almost optimal order in space a posteriori error bound in the l l 2 norm the interface is assumed to be of arbitrary shape but is of class c 2 for our purpose numerical results are presented to validate the derived estimators

three dimensional immersed finite element methods for electric field simulation in composite materials this paper presents two immersed finite element ife methods for solving the elliptic interface problem arising from electric field simulation in composite materials the meshes used in these ife methods can be independent of the interface geometry and position therefore if desired a structured mesh such as a cartesian mesh can be used in an ife method to simulate 3 d electric field in a domain with non trivial interfaces separating different materials numerical examples are provided to demonstrate that the accuracies of these ife methods are comparable to the standard linear finite element method with unstructured body fit mesh copyright 2005 john wiley sons ltd

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

an interface fitted mesh generator and virtual element methods for elliptic interface problems abstract a simple and efficient interface fitted mesh generation algorithm which can produce a semi structured interface fitted mesh in two and three dimensions quickly is developed in this paper elements in such interface fitted meshes are not restricted to simplices but can be polygons or polyhedra especially in 3d the polyhedra instead of tetrahedra can avoid slivers virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions algebraic multigrid solvers are used to solve the resulting linear algebraic system numerical results are presented to illustrate the effectiveness of our method

stability criteria of differential systems via liapunov s second method stability notions for systems of ordinary and functional differential equations were investigated by many authors our purpose in this paper is to study sufficient conditions of different types of stability and improvements of some results of athanassov yoneyama and li fu furthermore we show that the condition v t 0 0 is not sometimes a necessary condition for stability results

an improved immersed finite element particle in cell method for plasma simulation abstract the particle in cell pic method has been widely used for plasma simulation because of its noise reduction capability and moderate computational cost the immersed finite element ife method is efficient for solving interface problems on cartesian meshes which is desirable for the pic method the combination of these two methods provides an effective tool for plasma simulation with complex interface boundary this paper introduces an improved ife pic method that enhances the performance in both ife and pic aspects for the electric field solver we adopt the newly developed partially penalized ife method with enhanced accuracy for pic implementation we introduce a new interpolation technique to ensure the conservation of the charge numerical examples are provided to demonstrate the features of the improved ife pic method

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

enhanced functional evaluation for the finite element penalty method abstract we analyze the impact of using the penalty method on the estimation of a prescribed solution functional or quantity of interest qoi specifically we consider the use of penalty methods to enforce dirichlet boundary constraints focusing our attention on boundary fluxes as qois we propose an enhanced estimator of the boundary flux that includes a term involving the derivative of the flux with respect to the penalty parameter we show that the new estimator reduces the error arising from the use of the penalty method from o to o a well posed adjoint problem associated with the boundary flux is also proposed following an analysis of the penalty method errors in the enhanced flux estimator are then controlled using adjoint based techniques several numerical experiments are presented to demonstrate that the enhanced estimator in combination with adjoint error estimation allows one to efficiently control both the discretization and penalty errors in target qois

an analysis of a broken p_1 nonconforming finite element method for interface problems we study some numerical methods for solving a second order elliptic problem with interface we introduce an immersed finite element method based on the broken p 1 nonconforming piecewise linear polynomials on interface triangular elements having edge averages as degrees of freedom these linear polynomials are broken to match the homogeneous jump condition along the interface which is allowed to cut through the element we prove optimal orders of convergence in the h 1 and l 2 norm next we propose a mixed finite volume method in the context introduced in s h chou d y kwak and k y kim math comp 72 2003 pp 525 539 using the raviart thomas mixed finite element and this broken p 1 nonconforming element the advantage of this mixed finite volume method is that once we solve the symmetric positive definite pressure equation without lagrangian multiplier the velocity can be computed locally by a simple formula this procedure avoids solving the saddle point problem furthermore we show optimal error estimates of velocity and pressure in our mixed finite volume method numerical results show optimal orders of error in the l 2 norm and broken h 1 norm for the pressure and in the h mathrm div norm for the velocity

an unfitted finite element method using discontinuous galerkin in this paper we present a new approach to simulations on complex shaped domains the method is based on a discontinuous galerkin dg method using trial and test functions defined on a structured grid essential boundary conditions are imposed weakly via the dg formulation this method offers a discretization where the number of unknowns is independent of the complexity of the domain we will show numerical computations for an elliptic scalar model problem in 2 and 3 convergence rates for different polynomial degrees are studied

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

stabilization mechanisms in discontinuous galerkin finite element methods in this paper we propose a new general framework for dg methods which allows to uncover a basic mechanism that ensures suitable stability properties of the methods we show that such a mechanism is essentially the same for apparently unrelated stabilizations such as jump penalty upwinding and hughes franca type residual based stabilizations

interior penalty preconditioners for mixed finite element approximations of elliptic problems it is established that an interior penalty method applied to second order elliptic problems gives rise to a local operator which is spectrally equivalent to the corresponding nonlocal operator arising from the mixed finite element method this relation can be utilized in order to construct preconditioners for the discrete mixed system as an example a family of additive schwarz preconditioners for these systems is constructed numerical examples which confirm the theoretical results are also presented

nonconforming immersed finite element spaces for elliptic interface problems abstract in this paper we use a unified framework introduced in chen and zou 1998 to study two nonconforming immersed finite element ife spaces with integral value degrees of freedom the shape functions on interface elements are piecewise polynomials defined on sub elements separated either by the actual interface or its line approximation in this unified framework we use the invertibility of the well known sherman morison systems to prove the existence and uniqueness of ife shape functions on each interface element in either a rectangular or triangular mesh furthermore we develop a multi edge expansion for piecewise functions and a group of identities for nonconforming ife functions which enable us to show the optimal approximation capability of these ife spaces

an augmented immersed finite element method for variable coefficient elliptic interface problems in two and three dimensions abstract an augmented immersed finite element method is proposed for solving elliptic interface problems with discontinuous and variable coefficients by introducing the jump of normal derivative of the solution along the interface as an augmented variable the interface problem is solved iteratively on cartesian meshes using the generalized minimal residual gmres iterative method within each iteration we need to solve a system of linear equations with the same symmetric and positive definite matrix but with different right hand sides the sequence of linear equations is solved efficiently by the multi grid solvers since the set up of coarse grid matrices and decompositions is only computed once extensive numerical examples show that the number of gmres iterations is nearly independent of the mesh size when a diagonal like preconditioning is applied thus the computational work of the proposed method is linearly proportional to the number of grid nodes implementation details are given and extensive numerical examples in two and three dimensions are provided to show the accuracy and efficiency of the proposed method

an iterative immersed finite element method for an electric potential interface problem based on given surface electric quantity interface problems involving the non homogeneous flux jump condition are critical for engineering designs in the magnetostatic electrostatic field in applications such as plasma simulation we often only know the total electric quantity on the surface of the object not the charge density distribution on the surface which appears as the non homogeneous flux jump condition in the usual interface problems considered in the literature for the magnetostatic electrostatic field based on structured meshes independent of the interface this article proposes an iterative method that employs both the immersed finite element ife method with non homogeneous flux jump conditions and the regular finite element method with ghost nodes introduced in the object to solve the 2d interface problem for the potential field according to the given total electric quantity on the surface of the object numerical experiments are provided to illustrate the accuracy and efficiency of the proposed method

higher degree immersed finite element spaces constructed according to the actual interface abstract we discuss the construction of higher degree immersed finite element ife spaces that can be used to solve two dimensional second order elliptic interface problems having general interfaces without requiring the mesh to be aligned with the material interfaces the optimal approximation capability of the proposed piecewise p th degree ife spaces are demonstrated by numerical experiments with interpolations numerical solutions to interface problems generated from a partially penalized method based on the proposed higher order ife spaces also suggest optimal convergence in both the l 2 and h 1 norms under mesh refinement

linear and bilinear immersed finite elements for planar elasticity interface problems this article is to discuss the linear which was proposed in 18 19 and bilinear immersed finite element ife methods for solving planar elasticity interface problems with structured cartesian meshes basic features of linear and bilinear ife functions including the unisolvent property will be discussed while both methods have comparable accuracy the bilinear ife method requires less time for assembling its algebraic system our analysis further indicates that the bilinear ife functions are guaranteed to be applicable to a larger class of elasticity interface problems than linear ife functions numerical examples are provided to demonstrate that both linear and bilinear ife spaces have the optimal approximation capability and that numerical solutions produced by a galerkin method with these ife functions for elasticity interface problem also converge optimally in both l 2 and semi h 1 norms

an immersed weak galerkin method for elliptic interface problems abstract in this paper we present an immersed weak galerkin method for solving second order elliptic interface problems the proposed method does not require solution meshes to be aligned with the interface consequently uniform cartesian meshes can be used for nontrivial interfacial geometry we show the existence and uniqueness of the numerical algorithm and provide the error analysis in the energy norm numerical results are reported to demonstrate the performance of the method

stable generalized finite element method sgfem for parabolic interface problems abstract we study a stable generalized finite element method sgfem for parabolic interface problems with smooth interfaces the standard finite element fe space based on a simple mesh is enriched by a one side distance function to simulate gradient jumps of exact solutions on the interfaces the mesh is quasi uniform and independent of the interfaces so that mesh matching or refinement along the interfaces is not needed the main idea to derive the sgfem is to modify the one side distance function by subtracting its fe interpolant the sgfem approximates the parabolic interface problem with an optimal convergence rate and is stable in that its scaled condition number is of the same order as the fem o h 2 h is a mesh size parameter and does not depend on the relative positions of the mesh and interfaces the proposed sgfem is a conforming method and is free from any penalty parameters or stability schemes moreover the components of the approximation functions of the sgfem are independent when extended to vector valued interface problems the optimal convergence rates of sgfem for curved interfaces are proven theoretically for the backward euler and crank nicolson formulations the convergence and conditioning are tested by numerical experiments

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

high degree discontinuous petrov galerkin immersed finite element methods using fictitious elements for elliptic interface problems abstract we propose a new strategy for constructing the p th degree immersed finite element ife spaces by applying the least squares framework in adjerid et al 2017 on fictitious elements this new construction method significantly reduces the ill conditioning caused by the small subelement issue in adjerid et al 2017 of solving the local ife shape functions the proposed ife spaces are employed in a discontinuous petrov galerkin dpg scheme to solve the second order elliptic interface problems we present a group of numerical examples to show that the dpgife method with the new p th degree ife space as the trial function space has the optimal convergence rate which improves the numerical results reported in adjerid et al 2017

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

critical initial imperfection of structures abstract a method is introduced for determining the critical initial imperfection of discretized structures that decreases the load bearing capacity most rapidly the effects of imperfections on simple critical points such as limit points of loads and simple bifurcation points are theoretically investigated based on the idea of the lyapunov schmidt decomposition developed in bifurcation theory imperfection sensitivity varies with the types of points nonetheless critical imperfection pattern is expressed in the same formula regardless of the types among various imperfections the most influential can be found in a quantitative manner the validity and the usability of the proposed method are illustrated through its application to simple example structures

discontinuous bubble scheme for elliptic problems with jumps in the solution abstract we propose a new numerical method to solve an elliptic problem with jumps both in the solution and derivative along an interface by considering a suitable function which has the same jumps as the solution we transform the problem into one without jumps then we apply the immersed finite element method in which we allow uniform meshes so that the interface may cut through elements to discretize the problem as introduced in 1 2 3 some convenient way of approximating the jumps of the solution by piecewise linear functions is suggested our method can also handle the case when the interface passes through grid points we believe this paper presents the first resolution of such cases numerical experiments for various problems show second order convergence in l2 and first order in h1 norms moreover the convergence order is very robust for all problems tested

critical imperfection of symmetric structures a method is presented to determine the critical most unfavorable initial imperfection of structures of regular polygonal symmetry denoted by dihedral groups a critical point of such structures is either simple or forced to be double by the symmetry and the group representation theory is employed to deal with the degeneracy due to symmetry the method is developed further to incorporate the symmetry of imperfections in many cases of practical interest the critical imperfection is given explicitly as the product of imperfection sensitivity matrix and the critical eigenvector it is shown that the symmetry in the critical eigenvectors of the tangent stiffness matrix is inherited to the symmetry in the critical imperfection mode