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This article is about the error analysis for a partially penalized immersed finite element (PPIFE) method designed to solve linear planar-elasticity problems whose Lamé parameters are piecewise constants with an interface-independent mesh. The bilinear form in this method contains penalties to handle the discontinuity in the global immersed finite element (IFE) functions across interface edges. We establish a stress trace inequality for IFE functions on interface elements, we employ a patch idea to derive an optimal error bound for the stress of the IFE interpolation on interface edges, and we design a suitable energy norm by which the bilinear form in this PPIFE method is coercive. These key ingredients enable us to prove that this PPIFE method converges optimally in both an energy norm and the usualL2norm under the standard piecewiseH2-regularity assumption for the exact solution. Features of the proposed PPIFE method are demonstrated with numerical examples.

We present a discontinuous immersed finite element (IFE) method for incompressible interfacial flows that are governed by the Stokes equations. The method is based on a Cartesian mesh with elements cut by the moving interface. On this fixed unfitted mesh, we employ an immersed Q 1 ∕ Q 0 finite element space constructed according to the location of the interface and pertinent interface jump conditions. As such, the smearing of solution across the interface is greatly reduced. The interface, represented by a sequence of marker points, is advected on the fixed background mesh by the local fluid velocity. The mesh is locally refined near the interface to further improve accuracy. Compared with the phase-field method on adaptive meshes, our method can achieve the same level of accuracy with much less degrees of freedoms. We present some numerical examples to validate and demonstrate the capability of the proposed method.

In this article, we present and analyze a weak Galerkin finite element method for stationary Navier–Stokes problems. This weak Galerkin finite element scheme is based on a shape regular partition consisting of arbitrary polygons/polyhedra. We first establish a discrete embedding inequality that is useful in weak Galerkin finite element analysis for nonlinear problems. Then the stability and unique existence are proved for the discrete velocity and pressure by means of a discrete inf–sup condition. Furthermore, we derive the optimal error estimates for velocity approximation in the discrete H 1 -norm and pressure approximation in the L 2 -norm, respectively. Numerical examples are provided that corroborate the optimal convergence of the proposed method.

This article reports our explorations for solving interface problems of the Helmholtz equation by immersed finite elements (IFE) on interface independent meshes. Two IFE methods are investigated: the partially penalized IFE (PPIFE) and discontinuous Galerkin IFE (DGIFE) methods. Optimal convergence rates are observed for these IFE methods once the mesh size is smaller than the optimal mesh size which is mainly dictated by the wave number. Numerical experiments also suggest that higher degree IFE methods are advantageous because of their larger optimal mesh size and higher convergence rates.

We construct and analyze a group of immersed finite element (IFE) spaces formed by linear, bilinear and rotated Q1 polynomials for solving planar elasticity equation involving interface. The shape functions in these IFE spaces are constructed through a group of approximate jump conditions such that the unisolvence of the bilinear and rotated Q1 IFE shape functions are always guaranteed regardless of the Lam\`e parameters and the interface location. The boundedness property and a group of identities of the proposed IFE shape functions are established. A multi-point Taylor expansion is utilized to show the optimal approximation capabilities for the proposed IFE spaces through the Lagrange type interpolation operators.

This article develops and analyzes a $p$th degree immersed finite element (IFE) method for solving the elliptic interface problems with meshes independent of the coefficient discontinuity in the in...

A new immersed finite element (IFE) method is developed for second-order elliptic problems with discontinuous diffusion coefficient. The IFE space is constructed based on the rotated- $$Q_1$$ nonconforming finite elements with the integral-value degrees of freedom. The standard nonconforming Galerkin method is employed in this IFE method without any stabilization term. Error estimates in energy and $$L^2$$ -norms are proved to be better than $$O(h\sqrt{|\log h|})$$ and $$O(h^2|\log h|)$$ , respectively, where the $$|\log h|$$ factors reflect jump discontinuity. Numerical results are reported to confirm our analysis.

The issue of tunnel face stability traversing the fault fracture zone with high hydraulic pressure has been addressed based on a new three-dimensional limit equilibrium model. First, the finite element numerical simulation is conducted to reveal the effects of the differences in permeability coefficient of strata on both sides of the geological interface on the distribution of seepage flow field nearby tunnel face. With this discovery, an existing universal approximate formula for hydraulic head nearby tunnel face is modified to accurately calculate the hydraulic head distributions under this scenario. Second, introducing the effect of geological interface, a new limit equilibrium model that consists of a trapezoid and a prism is proposed. The distributed loading exerted by the prism upon the trapezoid is deduced by considering the effects of seepage force and geological interface. Then, limit support pressure is derived according to the limit equilibrium state of the trapezoid. Finally, the comparison of the predicting results of the proposed limit equilibrium model with the results of the numerical simulations and the existing failure mechanism is performed, which indicates that the proposed trapezoid-prism model can well predict the tunnel face stability traversing the fault fracture zone with high hydraulic pressure.

A group of partially penalized immersed finite element (PPIFE) methods for second-order hyperbolic interface problems were discussed in Yang (Numer Math Theor Methods Appl 11:272–298, 2018) where the author proved their optimal O(h) convergence in an energy norm under a sub-optimal piecewise $$H^3$$ regularity assumption. In this article, we reanalyze the fully discrete PPIFE method presented in Yang (2018). Utilizing the error bounds given recently in Guo et al. (Int J Numer Anal Model 16(4):575–589, 2019) for elliptic interface problems, we are able to derive optimal a-priori error bounds for this PPIFE method not only in the energy norm but also in $$L^2$$ norm under the standard piecewise $$H^2$$ regularity assumption in the space variable of the exact solution, rather than the excessive piecewise $$H^3$$ regularity. Numerical simulations for standing and travelling waves are presented, which corroboratively confirm the reported error analysis.

We study the weak Galerkin finite element method for stationary Navier–Stokes problem. We propose a weak finite element velocity-pressure space pair that satisfies the discrete inf-sup condition. This space pair is then employed to construct a stable weak Galerkin finite element scheme without adding any stabilizing term or penalty term. We prove a discrete embedding inequality on the weak finite element space which, together with the discrete inf-sup condition, enables us to establish the unique existence and stability estimates of the discrete velocity and pressure. Then, we derive the optimal error estimates for velocity and pressure approximations in the H 1 -norm and L 2 -norm, respectively. Numerical experiments are provided to illustrate the theoretical analysis.

Surface evolution is an unavoidable issue in engineering plasma applications. In this article an iterative method for modeling plasma-surface interactions with moving interface is proposed and validated. In this method, the plasma dynamics is simulated by an immersed finite element particle-in-cell (IFE-PIC) method, and the surface evolution is modeled by the Huygens wavelet method which is coupled with the iteration of the IFE-PIC method. Numerical experiments, including prototypical engineering applications, such as the erosion of Hall thruster channel wall, are presented to demonstrate features of this Huygens IFE-PIC method for simulating the dynamic plasma-surface interactions.

We study the weak Galerkin finite element method for Stokes problem. A new weak Galerkin finite element velocity-pressure space pair is presented which satisfies the discrete inf-sup condition. Based on this space pair, we establish a stable weak Galerkin approximation scheme without adding any stability term or penalty term. Then, we further derive the optimal error estimates for velocity and pressure approximations, respectively. Numerical experiments are provided to illustrate the theoretical analysis., 22pages

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.

We present a unified framework for developing and analysing immersed finite element (IFE) spaces for solving typical elliptic interface problems with interface independent meshes. This framework allows us to construct a group of new IFE spaces with either linear, or bilinear, or the rotated-Q1 polynomials. Functions in these IFE spaces are locally piecewise polynomials defined according to the sub-elements formed by the interface itself instead of its line approximation. We show that the unisolvence for these IFE spaces follows from the invertibility of the Sherman-Morrison matrix. A group of estimates and identities are established for the interface geometry and shape functions that are applicable to all of these IFE spaces. Most importantly, these fundamental preparations enable us to develop a unified multipoint Taylor expansion procedure for proving that these IFE spaces have the expected optimal approximation capability according to the involved polynomials., Comment: 27 pages, 3 figures

This article presents an error analysis of the symmetric linear/bilinear partially penalized immersed finite element (PPIFE) methods for interface problems of Helmholtz equations. Under the assumption that the exact solution possesses a usual piecewise $H^2$ regularity, the optimal error bounds for the PPIFE solutions are derived in an energy norm and the usual $L^2$ norm provided that the mesh size is sufficiently small. A numerical example is conducted to validate the theoretical conclusions., 14 pages, 1 figure

Motivated by the need to handle complex boundary conditions efficiently and accurately in particle-in-cell (PIC) simulations, this paper presents a three-dimensional (3D) linear immersed finite element (IFE) method with non-homogeneous flux jump conditions for solving electrostatic field involving complex boundary conditions using structured meshes independent of the interface. This method treats an object boundary as part of the simulation domain and solves the electric field at the boundary as an interface problem. In order to resolve charging on a dielectric surface, a new 3D linear IFE basis function is designed for each interface element to capture the electric field jump on the interface. Numerical experiments are provided to demonstrate the optimal convergence rates in L 2 and H 1 norms of the IFE solution. This new IFE method is integrated into a PIC method for simulations involving charging of a complex dielectric surface in a plasma. A numerical study of plasma-surface interactions at the lunar terminator is presented to demonstrate the applicability of the new method.

This article presents an immersed finite element (IFE) method for solving the typical three-dimensional second order elliptic interface problem with an interface-independent Cartesian mesh. The local IFE space on each interface element consists of piecewise trilinear polynomials constructed by extending polynomials from one subelement to the whole element according to the jump conditions of the interface problem. In this space, the IFE shape functions with the Lagrange degrees of freedom can always be constructed regardless of interface location and discontinuous coefficients. The proposed IFE space is proven to have the optimal approximation capabilities to the functions satisfying the jump conditions. A group of numerical examples with representative interface geometries are presented to demonstrate features of the proposed IFE method.

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.

This article presents a finite element method on a fixed mesh for solving a group of inverse geometric problems for recovering the material interface of a linear elasticity system. A partially penalized immersed finite element method is used to discretize both the elasticity interface problems and the objective shape functionals accurately regardless of the shape and location of the interface. Explicit formulas for both the velocity fields and the shape derivatives of IFE shape functions are derived on a fixed mesh and they are employed in the shape sensitivity framework through the discretized adjoint method for accurately and efficiently computing the gradients of objective shape functions with respect to the parameters of the interface curve. The shape optimization for solving an inverse geometric problem is therefore accurately reduced to a constrained optimization that can be implemented efficiently within the IFE framework together with a standard optimization algorithm. We demonstrate features and advantages of the proposed IFE-based shape optimization method by several typical inverse geometric problems for linear elasticity systems.

Tunnel excavation inevitably causes variations in stress and displacement fields in soil, which could affect the serviceability and safety of the adjacent underground structures. Change of the earth pressure acting on the existing structures is closely related to soil arching effect caused by a new tunnel excavation, but it is still lack of full understanding. In this study, a finite element method (FEM) model was first verified by field measurements based on a case history of Metro Line 2 construction in Changsha, China. Then, a series of simplified three-dimensional (3D) FEM models were established to investigate the stress redistribution and the soil arching evolution induced by earth pressure balance shield (EPBS) tunneling. The changes of the earth pressure, the coefficient of lateral earth pressure, and the settlement of soil mass above the tunnel during tunneling were analyzed and the soil arching zone was determined. 3D stress-transfer mechanism in soil arching zone during tunneling was revealed. In addition, the influences of shield-driving parameters (i.e. support pressure, grouting pressure) on soil arching evolution were also investigated. The results show that the loosened zone extends to nearly 0.73D (D = tunnel diameter) above the tunnel crown in vertical direction. The height of the arch zone above the loosened zone is about 1.27D. The horizontal soil arching in front of EPBS occurs near the ground surface. With EPBS advancing, the horizontal soil arching gradually transforms into the vertical soil arching. After the installation of lining, the earth pressure is finally mainly transferred by vertical soil arching in the transverse section. The proper combination of shield-driving parameters for controlling the expansion of the loosened zone is support pressure of 0.6P1 − 2.2P1 and grouting pressure of 1.0P2 − 1.8P2, where P1 is initial horizontal stress at the tunnel axis and P2 = 1.2P1.

We present a new fixed mesh algorithm for solving a class of interface inverse problems for the typical elliptic interface problems. These interface inverse problems are formulated as shape optimization prob- lems whose objective functionals depend on the shape of the interface. Regardless of the location of the interface, both the governing partial differential equations and the objective functional are discretized optimally, with respect to the involved polynomial space, by an immersed finite element (IFE) method on a fixed mesh. Furthermore, the formula for the gradient of the descritized objective function is de- rived within the IFE framework that can be computed accurately and efficiently through the discretized adjoint procedure. Features of this proposed IFE method based on a fixed mesh are demonstrated by its applications to three representative interface inverse problems: the interface inverse problem with an internal measurement on a sub-domain, a Dirichlet-Neumann type inverse problem whose data is given on the boundary, and a heat dissipation design problem.

We present a discontinuous immersed finite element (IFE) method for Stokes interface problems on Cartesian meshes that do not require the mesh to be aligned with the interface. As such, the method allows unfitted meshes with elements cut by the interface and thus, may contain more than one fluid. On these unfitted meshes we construct an immersed Q 1 / Q 0 finite element space according to the location of the interface and pertinent interface jump conditions. The proposed Q 1 / Q 0 IFE shape functions have several desirable features such as the unisolvence and the partition of unity. We present several numerical examples to demonstrate that the proposed IFE spaces maintain the optimal approximation capability with respect to the polynomials used. We also show that related discontinuous IFE solutions of Stokes interface problems maintain the optimal convergence rates in both L 2 and broken H 1 norms.

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 these IFE methods can be proven to have the optimal convergence rate in the H1-norm provided that the exact solution has sufficient regularity, but also numerical results indicate that their convergence rates in both the H1-norm and the L2-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 also are of a great potential to be useful in error analysis for other IFE methods.

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.

Many researchers applied creepage to calculating creep force, but relation between creepage and creep force are very complex and it has been a major problem for solving rolling contact theory. The paper uses the finite element method for studying directly creep forces and apply it to the studies of wheel-rail rolling contact. Paper establishes the finite element model of wheel-rail rolling contact based on ALE method in ABAQUS, and simulated unsteady state of wheel-rail rolling contact and research normal force of wheel-rail contact pitch. Results show that the normal maximum pressure in left wheel have a trend toward the center of contact pitch and the normal maximum pressure in right wheel have a trend toward the other center of contact pitch with increasement of displacement

Usually the gearbox casing of twin screw pulping extruder was designed by experience and greater weight. In order to reduce the casing weight under the premise of ensuring its stiffness and strength, firstly the casting lower box and the weakest axle in the casing were analyzed with finite element method by ANSYS software. And then the optimum design was carried out to take the minimum volume as an objective function. Finally, the welding casing has been designed as considering the manufacturing cost. The results show that the weight of the casting lower box after optimization is 22.95% lower than that of the experience design, and the weight of the welding part is 19.09% lower than that. The work has certain significance for optimum design of gearbox casing.

Many researchers applied creepage to calculating creep force, and relation between creepage and creep force are very complex and it has been a major problem for solving rolling contact theory. The paper established the finite element model of wheel-rail rolling contact based on ALE method (Arbitrary Lagrange-Euler method) in ABAQUS, and research impact of yaw motion of wheelset on creepage of wheel-rail contact. Results show that in the yaw movement there are small fluctuations in elastic longitudinal creepage; elastic lateral wheel creepage of left and right wheel appear different, there are small change after yaw angle beyond 2 mrad; spin creepage of left and right wheel have not change with increase of yaw angle

Many researchers applied creepage to calculating creep force, and relation between creepage and creep force are very complex and it has been a major problem for solving rolling contact theory. The paper established the finite element model of wheel-rail rolling contact based on ALE method (Arbitrary Lagrange-Euler method) in ABAQUS, and research impact of lateral motion of wheelset on creepage of wheel-rail contact. Results show in the lateral movement value difference between their longitudinal creepage is small, and maximum error is less than 20%, there is more distinct change trend about lateral creepage in left contact area than one in right contact area, and spin creepage of left wheel drop down in 4mm lateral displacement.

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.

Finite element model which can reflect the effects of different loading positions were constructed with Beam 188. Effects of different restraints, load models and rise-span ratios on out-of-plane buckling were studied by comparing results of fixed arches with that of pin-end arches under three loading models. It is conservative to design by employing results of radial loading. For ideal restraints, out-of-plane stability of pin-end arches is better than fixed arches when rise-span ratio is big. Effects of different loading positions on out-of-plane buckling were studied. Buckling loads of arches which are loaded at arch-axises are bigger than those of top flanges, but smaller than those of bottom flanges.

This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time. The method presented here uses immersed finite element (IFE) functions for the discretization in spatial variables that can be carried out over a fixed mesh (such as a Cartesian mesh if desired), and this feature makes it possible to reduce the parabolic equation to a system of ordinary differential equations (ODE) through the usual semi-discretization procedure. Therefore, with a suitable choice of the ODE solver, this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured (Cartesian) mesh. Numerical examples are presented to demonstrate features of this new method.

This article proposes a nonconforming immersed finite element (IFE) method for solving planar elasticity interface problems with structured (or Cartesian) meshes even if the material interface has a nontrivial geometry. IFE functions developed in this article are applicable to arbitrary configurations of elasticity materials and interface locations. Optimal approximation capability is observed for this new IFE space. The displacement Galerkin method based on this IFE space is robust (locking-free). Numerical experiments are presented to demonstrate that the IFE solution converges optimally for both compressible and nearly incompressible materials.

This article proposes a selective immersed discontinuous Galerkin method based on bilinear immersed finite elements (IFE) for solving second-order elliptic interface problems. This method applies the discontinuous Galerkin formulation wherever selected, such as those elements around an interface or a singular source, but the regular Galerkin formulation everywhere else. A selective bilinear IFE space is constructed and applied to the selective immersed discontinuous Galerkin method based on either the symmetric or nonsymmetric interior penalty discontinuous Galerkin formulation. The new method can solve an interface problem by a rectangular mesh with local mesh refinement independent of the interface even if its geometry is nontrivial. Meanwhile, if desired, its computational cost can be maintained very close to that of the standard Galerkin IFE method. It is shown that the selective bilinear IFE space has the optimal approximation capability expected from piecewise bilinear polynomials. Numerical examples are provided to demonstrate features of this method, including the effectiveness of local mesh refinement around the interface and the sensitivity to the penalty parameters. Copyright © 2013 John Wiley & Sons, Ltd.

Introduction The aim of this study was to explore the biomechanical effects on the craniomaxillary complex of bone anchorage and dental anchorage during maxillary protraction. Methods We established 2 finite element models. One simulated maxillary protraction with dental anchorage in the maxillary first molars and the other with bone anchorage in the infrazygomatic buttresses of the maxilla. The magnitude of the applied forces was 500 g per side, and the force directions were 0°, 10°, 20°, and 30° forward and downward relative to the occlusal plane. Results The finite element model of the craniomaxillary complex could displace in an almost translatory manner when the force direction was about 20° in the bone anchorage model and about 30° in the dental anchorage model. The nodes representing the sutures at the back of the maxilla showed greater stress in the bone anchorage model than in the dental anchorage model in the same force direction. It is the opposite at the front of the maxilla. Conclusions We should determine the direction of applied force according to the anchorage location and skeletal characteristics of patients before maxillary protraction. The dramatic effects of maxillary protraction with bone anchorage can be based on the advantages of bone anchorage, not on the changes in the region of the applied force.

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.

This paper applies bilinear immersed finite elements (IFEs) in the interior penalty discontinuous Galerkin (DG) methods for solving a second order elliptic equation with discontinuous coefficient. A discontinuous bilinear IFE space is constructed and applied to both the symmetric and nonsymmetric interior penalty DG formulations. The new methods can solve an interface problem on a Cartesian mesh independent of the interface with local refinement at any locations needed even if the interface has a nontrivial geometry. Numerical examples are provided to show features of these methods.

In this manuscript we present a p-th degree immersed finite element method for solving boundary value problems with discontinuous coefficients. In this method, interface jump conditions are employed in the finite element basis functions, and the mesh does not have to be aligned with coefficient discontinuity. We show that under h refinement the immersed finite element solution converges to the true solution at the optimal O(h^p^+^1) and O(h^p) rates in the L^2 and H^1 norms, respectively. Furthermore, numerical results suggest that the immersed finite element solution converges exponentially fast under p refinement. Numerical examples are provided to illustrate features of this immersed finite element method.

This article discusses a bilinear immersed finite element (IFE) space for solving second-order elliptic boundary value problems with discontinuous coefficients (interface problem). This is a nonconforming finite element space and its partition can be independent of the interface. The error estimates for the interpolation of a Sobolev function indicate that this IFE space has the usual approximation capability expected from bilinear polynomials. Numerical examples of the related finite element method are provided. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008

This paper discusses a class of quadratic immersed finite element (IFE) spaces developed for solving second order elliptic interface problems. Unlike the linear IFE basis functions, the quadratic IFE local nodal basis functions cannot be uniquely defined by nodal values and interface jump conditions. Three types of one dimensional quadratic IFE basis functions are presented together with their extensions for forming the two dimensional IFE spaces based on rectangular partitions. Approximation capabilities of these IFE spaces are discussed. Finite element solutions based on these IFE for representative interface problems are presented to further illustrate capabilities of these IFE spaces.

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.

This article discusses an immersed finite element (IFE) space introduced for solving a second-order elliptic boundary value problem with discontinuous coefficients (interface problem). The IFE space is nonconforming and its partition can be independent of the interface. The error estimates for the interpolation of a function in the usual Sobolev space indicate that this IFE space has an approximation capability similar to that of the standard conforming linear finite element space based on body-fit partitions. Numerical examples of the related finite element method based on this IFE space are provided. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 338–367, 2004

In this paper we are concerned with finite element approximations to a nonlinear elliptic partial differential equation with homogeneous Dirichlet boundary conditions. This kind of problems arises for example from modeling a stationary heat conduction in nonlinear inhomogeneous and anisotropic media. For finite elements of degree k \geq 1 in each variable, by means of an interpolation postprocessing technique, we obtain the global superconvergence of O(hk + 1) in the H1-norm and O(hk + 2) in the L2-norm provided the weak solution is sufficiently smooth. As by-products, the global superconvergence results can be used to generate efficient a posteriori error estimators. Representative numerical examples are also given to illustrate our theoretical analysis.