A second order isoparametric finite element method for elliptic interface problems

A second order isoparametric finite element method (IPFEM) is proposed for elliptic interface problems. It yields better accuracy than some existing second-order methods, when the coefficients or the flux across the immersed curved interface is discontinuous. Based on an initial Cartesian mesh, a mesh optimization strategy is presented by employing curved boundary elements at the interface, and an incomplete quadratic finite element space is constructed on the optimized mesh. It turns out that the number of curved boundary elements is far less than that of the straight one, and the total degree of freedom is almost the same as the uniform Cartesian mesh. Numerical examples with simple and complicated geometrical interfaces demonstrate the efficiency of the proposed method.     

Superconvergence of immersed finite element methods for interface problems

In this article, we study superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that immersed finite element solutions inherit all desired superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. In particular, on interface elements, superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and interface jump conditions.     

基于Crouzeix-Raviart元的界面浸入有限元方法及其收敛性分析

本文对具间断系数的二阶椭圆界面问题提出一种浸入有限元方法(theimmersed finite element method), 即在界面单元上采用依赖于界面的线性多项式空间离散, 而在非界面单元上采用Crouzeix-Raviart非协调元离散. 论证表明, 该方法具有对界面问题解的最优L²-模和H¹-模收敛精度.     

A new augmented immersed finite element method without using SVD interpolations

Augmented immersed interface methods have been developed recently for interface problems and problems on irregular domains including CFD applications with free boundaries and moving interfaces. In an augmented method, one or several augmented variables are introduced along the interface or boundary so that one can get efficient discretizations. The augmented variables should be chosen such that the interface or boundary conditions are satisfied. The key to the success of the augmented methods often relies on the interpolation scheme to couple the augmented variables with the governing differential equations through the interface or boundary conditions. This has been done using a least squares interpolation (under-determined) for which the singular value decomposition (SVD) is used to solve for the interpolation coefficients. In this paper, based on properties of the finite element method, a new augmented immersed finite element method (IFEM) that does not need the interpolations is proposed for elliptic interface problems that have a piecewise constant coefficient. Thus the new augmented method is more efficient and simple than the old one that uses interpolations. The method then is extended to Poisson equations on irregular domains with a Dirichlet boundary condition. Numerical experiments with arbitrary interfaces/irregular domains and large jump ratios are provided to demonstrate the accuracy and the efficiency of the new augmented methods. Numerical results also show that the number of GMRES iterations is independent of the mesh size and nearly independent of the jump in the coefficient.     

A uniformly well-conditioned, unfitted Nitsche method for interface problems

A finite element method for elliptic partial differential equations that allows for discontinuities along an interface not aligned with the mesh is presented. The solution on each side of the interface is separately expanded in standard continuous, piecewise-linear functions, and jump conditions at the interface are weakly enforced using a variant of Nitsche’s method. In our method, the solutions on each side of the interface are extended to the entire domain which results in a fixed number of unknowns independent of the location of the interface. A stabilization procedure is included to ensure well-defined extensions. We prove that the method provides optimal convergence order in the energy and the L ² norms and a condition number of the system matrix that is independent of the position of the interface relative to the mesh. Numerical experiments confirm the theoretical results and demonstrate optimal convergence order also for the pointwise errors.     

四边形剖分下的组合多尺度有限元方法求解多尺度椭圆问题（英文）

The combined finite element and multiscale finite element method(FEMsFEM) [W. Deng and H. Wu, Multiscale Model. Simul., 12(2014), pp.1424-1457.]has been introduced for the multiscale elliptic problems. This is accomplished by using the standard finite element method on a fine mesh of the problematic part of the domain and using the oversampling MsFEM on a coarse mesh of the other part. The transmission condition across the FE-MsFE interface is treated by the penalty technique. FE-MsFEM can solve the multiscale elliptic problems with fine and long-ranged high contrast channels very efficiently. However, the detailed convergence analysis reveals that the error generated by the mismatch between the triangulation and the period of the coefficient still exists. A direct approach to reduce this error is to utilize the rectangle mesh for the domain. In this paper,we investigate the FE-MsFEM based on the rectangle mesh for the multiscale elliptic problems. Error estimate is given under the assumption that the oscillating coefficient is periodic. Numerical experiments for the rectangle mesh are carried out on the multiscale problems with periodic highly oscillating coefficient and high contrast channels. Their results demonstrate the efficiency of the proposed method.     

PPIFE Method with Non-Homogeneous Flux Jump Conditions and Its Efficient Numerical Solver for Elliptic Optimal Control Problems with Interfaces

In this paper, we design a partially penalized immersed finite element method for solving elliptic interface problems with non-homogeneous flux jump conditions. The method presented here has the same global degrees of freedom as classic immersed finite element method. The non-homogeneous flux jump conditions can be handled accurately by additional immersed finite element functions. Four numerical examples are provided to demonstrate the optimal convergence rates of the method in $L^{\infty}$, $L^{2}$ and $H^{1}$ norms. Furthermore, the method is combined with post-processing technique to solve elliptic optimal control problems with interfaces. To solve the resulting large-scale system, block diagonal preconditioners are introduced. These preconditioners can lead to fast convergence of the Krylov subspace methods such as GMRES and are independent of the mesh size. Four numerical examples are presented to illustrate the efficiency of the numerical schemes and preconditioners.     

A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.     

A partially penalty immersed interface finite element method for anisotropic elliptic interface problems

In this article, we develop a partially penalty immersed interface finite element (PIFE) method for a kind of anisotropy diffusion models governed by the elliptic interface problems with discontinuous tensor‐coefficients. This method is based on linear immersed interface finite elements (IIFE) and applies the discontinuous Galerkin formulation around the interface. We add two penalty terms to the general IIFE formulation along the sides intersected with the interface. The flux jump condition is weakly enforced on the smooth interface. By proving that the piecewise linear function on an interface element is uniquely determined by its values at the three vertices under some conditions, we construct the finite element spaces. Therefore, a PIFE procedure is proposed, which is based on the symmetric, nonsymmetric or incomplete interior penalty discontinuous Galerkin formulation. Then we prove the consistency and the solvability of the procedure. Theoretical analysis and numerical experiments show that the PIFE solution possesses optimal‐order error estimates in the energy norm and norm.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1984–2028, 2014     

Parallel multi-frontal solver for p adaptive finite element modeling of multi-physics computational problems

The paper presents a parallel direct solver for multi-physics problems. The solver is dedicated for solving problems resulting from adaptive finite element method computations. The concept of finite element is actually replaced by the concept of the node. The computational mesh consists of several nodes, related to element vertices, edges, faces and interiors. The ordering of unknowns in the solver is performed on the level of nodes. The concept of the node can be efficiently utilized in order to recognize unknowns that can be eliminated at a given node of the elimination tree. The solver is tested on the exemplary three-dimensional multi-physics problem involving the computations of the linear acoustics coupled with linear elasticity. The three-dimensional tetrahedral mesh generation and the solver algorithm are modeled by using graph grammar formalism. The execution time and the memory usage of the solver are compared with the MUMPS solver.     

Equivalent operator preconditioning for elliptic problems with nonhomogeneous mixed boundary conditions

The numerical solution of linear elliptic partial differential equations often involves finite element discretization, where the discretized system is usually solved by some conjugate gradient method. The crucial point in the solution of the obtained discretized system is a reliable preconditioning, that is to keep the condition number of the systems under control, no matter how the mesh parameter is chosen. The PCG method is applied to solving convection-diffusion equations with nonhomogeneous mixed boundary conditions. Using the approach of equivalent and compact-equivalent operators in Hilbert space, it is shown that for a wide class of elliptic problems the superlinear convergence of the obtained preconditioned CGM is mesh independent under FEM discretization.     

A mortar element method for elliptic problems with discontinuous coefficients

This paper proposes a mortar finite element method for solvingthe two-dimensional second-order elliptic problem with jumpsin coefficients across the interface between two subregions.Non-matching finite element grids are allowed on the interface,so independent triangulations can be used in different subregions.Explicitly realizable mortar conditions are introduced to couplethe individual discretizations. The same optimal L²-norm andenergy-norm error estimates as for regular problems are achievedwhen the interface is of arbitrary shape but smooth, thoughthe regularity of the true solution is low in the whole physicaldomain.     

A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints

In this paper optimal control problems governed by elliptic semilinear equations and subject to pointwise state constraints are considered. These problems are discretized using finite element methods and a posteriori error estimates are derived assessing the error with respect to the cost functional. These estimates are used to obtain quantitative information on the discretization error as well as for guiding an adaptive algorithm for local mesh refinement. Numerical examples illustrate the behavior of the method.     

Finite element methods for semilinear elliptic problems with smooth interfaces

The purpose of this paper is to study the finite element method for second order semilinear elliptic interface problems in two dimensional convex polygonal domains. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with straight interface triangles [Numer. Math., 79 (1998), pp. 175–202]. For a finite element discretization based on a mesh which involve the approximation of the interface, optimal order error estimates in L ² and H ¹-norms are proved for linear elliptic interface problem under practical regularity assumptions of the true solution. Then an extension to the semilinear problem is also considered and optimal error estimate in H ¹ norm is achieved.     

A Well-Conditioned,Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems

In this paper, we introduce a nonconforming Nitsche's extended finite element method (NXFEM) for elliptic interface problems on unfitted triangulation elements. The solution on each side of the interface is separately expanded in the standard nonconforming piecewise linear polynomials with the edge averages as degrees of freedom. The jump conditions on the interface and the discontinuities on the cut edges (the segment of edges cut by the interface) are weakly enforced by the Nitsche's approach. In the method, the harmonic weighted fluxes are used and the extra stabilization terms on the interface edges and cut edges are added to guarantee the stability and the well conditioning. We prove that the convergence order of the errors in energy and $L^2$ norms are optimal. Moreover, the errors are independent of the position of the interface relative to the mesh and the ratio of the discontinuous coefficients. Furthermore, we prove that the condition number of the system matrix is independent of the interface position. Numerical examples are given to confirm the theoretical results.     

An adaptive finite element method with asymptotic saturation for eigenvalue problems

This paper discusses adaptive finite element methods for the solution of elliptic eigenvalue problems associated with partial differential operators. An adaptive method based on nodal-patch refinement leads to an asymptotic error reduction property for the computed sequence of simple eigenvalues and eigenfunctions. This justifies the use of the proven saturation property for a class of reliable and efficient hierarchical a posteriori error estimators. Numerical experiments confirm that the saturation property is present even for very coarse meshes for many examples; in other cases the smallness assumption on the initial mesh may be severe.     

A hybrid coupling interface method for elliptic complex interface problems

We propose a coupling interface method (CIM) under Cartesian grid for solving elliptic complex interface problems in arbitrary d dimensions, where the coefficients, the source terms, and the solutions may be discontinuous or singular across the interfaces. It consists of a first-order version (CIM1) and a second-order version (CIM2). In one dimension, this finite difference method at a grid point adjacent to the interface is derived based on piecewise linear (CIM1) or quadratic (CIM2) approximation of the solution and two jump conditions. The method is extended to high dimensions through a dimensionby-dimension approach. To connect information from each dimension, a coupled equation for the principal derivatives is derived through the jump conditions in each coordinate direction. For CIM2, one-side interpolation for cross derivatives is need. This coupling approach reduces number of grid point in the finite difference stencil. The hybrid method uses CIM1 or CIM2 adaptly for complex interface. Numerical tests demonstrate that CIM1 and CIM2 are respectively first order and second order in the maximal norm with less error as compared with other methods. In addition, the hybrid CIM can solve complex interface problems in two and three dimensions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonconforming h-p spectral element methods for elliptic problems

In this paper we show that we can use a modified version of the h-p spectral element method proposed in [6,7,13,14] to solve elliptic problems with general boundary conditions to exponential accuracy on polygonal domains using nonconforming spectral element functions. A geometrical mesh is used in a neighbourhood of the corners. With this mesh we seek a solution which minimizes the sum of a weighted squared norm of the residuals in the partial differential equation and the squared norm of the residuals in the boundary conditions in fractional Sobolev spaces and enforce continuity by adding a term which measures the jump in the function and its derivatives at inter-element boundaries, in fractional Sobolev norms, to the functional being minimized. In the neighbourhood of the corners, modified polar coordinates are used and a global coordinate system elsewhere. A stability estimate is derived for the functional which is minimized based on the regularity estimate in [2]. We examine how to parallelize the method and show that the set of common boundary values consists of the values of the function at the corners of the polygonal domain. The method is faster than that proposed in [6,7,14] and the h-p finite element method and stronger error estimates are obtained.     

Consistency,stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems

Summary. In an abstract framework we present a formalism which specifies the notions of consistency and stability of Petrov-Galerkin methods used to approximate nonlinear problems which are, in many practical situations, strongly nonlinear elliptic problems. This formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems. This theory is illustrated with the example: in a two dimensional domain with Dirichlet boundary conditions. Received June 10, 1992 / Revised version received February 28, 1994     

A mortar element method for hyperbolic problems

A non-conforming finite element method based on non-overlapping domain decomposition is extended to linear hyperbolic problems. The method is based on streamline-diffusion/discontinuous Galerkin methods and the mortar element method. A weak flux continuity condition at the inflow interface is enforced by means of Lagrange multipliers. This weak flux continuity condition replaces the usual mortar condition for elliptic problems, and allows non-matching grids at the subdomain interfaces. To cite this article: Y. Bourgault, A. El Boukili, C. R. Acad. Sci. Paris, Ser. I 338 (2004).