Nitsche Type Mortaring for Singularly Perturbed Reaction-diffusion Problems

The paper is concerned with the Nitsche mortaring in the framework of domain decomposition where non-matching meshes and weak continuity of the finite element approximation at the interface are admitted. The approach is applied to singularly perturbed reaction-diffusion problems in 2D. Non-matching meshes of triangles being anisotropic in the boundary layers are applied. Some properties as well as error estimates of the Nitsche mortar finite element schemes are proved. In particular, using a suitable degree of anisotropy of triangles in the boundary layers of a rectangle, convergence rates as known for the conforming finite element method are derived. Numerical examples illustrate the approach and the results.     

The Fourier-Nitsche-mortaring for elliptic problems with reentrant edges

The Fourier method is combined with the Nitsche-finite-element method (as a mortar method) and applied to the Dirichlet problem of the Poisson equation in three-dimensional axisymmetric domains with reentrant edges generating singularities. The approximating Fourier method yields a splitting of the 3D problem into a set of 2D problems on the meridian plane of the given domain. For solving the 2D problems bearing corner singularities, the Nitsche-finite-element method with non-matching meshes and mesh grading near reentrant corners is applied. Using the explicit representation of some singularity function of non-tensor product type, the rate of convergence of the Fourier-Nitsche-mortaring is estimated in some H ¹-like norm as well as in the L ₂-norm for weak regularity of the solution. Finally, some numerical results are presented.      

An improved interface element with variable nodes for non-matching finite element meshes

Extensive improvements of the interface element method (IEM) are proposed for the efficient treatment of non-matching finite element meshes. Our approach enables us to establish the master element via the moving least-square (MLS) approximation, and so to remove the cumbersome process of constructing interface elements. The values of shape functions and their derivatives are therefore mapped from the master element, as in the conventional finite element method. For the assurance of convergence and compatibility condition, a patch test is demonstrated. Through several examples of 2D linear elasticity, the convergence rate is compared between the present interface element and the previous version.     

Non-Nested Multi-Level Solvers for Finite Element Discretisations of Mixed Problems

We consider a general framework for analysing the convergence of multi-grid solvers applied to finite element discretisations of mixed problems, both of conforming and nonconforming type. As a basic new feature, our approach allows to use different finite element discretisations on each level of the multi-grid hierarchy. Thus, in our multi-level approach, accurate higher order finite element discretisations can be combined with fast multi-level solvers based on lower order (nonconforming) finite element discretisations. This leads to the design of efficient multi-level solvers for higher order finite element discretisations. Received May 17, 2001; revised February 2, 2002 Published online April 25, 2002     

Stable Evaluation of Jacobian Matrices on Highly Refined Finite Element Meshes

In the adaptive finite element method, the solution of a p.d.e. is approximated by finer and finer meshes, which are controlled from error estimators. So, starting from a given coarse mesh, some elements are subdivided a couple of times. We investigate the question of avoiding instabilities which limit this process from the fact that nodal coordinates of one element coincide in more and more leading digits. To overcome this problem we demonstrate a simple mechanism for red subdivision of triangles (and hanging nodes) and a more sophisticated technique for general quadrilaterals.     

A Black-Box Iterative Solver Based on a Two-Level Schwarz Method

We propose a black-box parallel iterative method suitable for solving both elliptic and certain non-elliptic problems discretized on unstructured meshes. The method is analyzed in the case of the second order elliptic problems discretized on quasiuniform P1 and Q1 finite element meshes. The numerical experiments confirm the validity of the proved convegence estimate and show that the method can successfully be used for more difficult problems (e.g. plates, shells and Helmholtz equation in high-frequency domain.) Received: July 28, 1997; revised June 20, 1999     

A Multigrid Method for Nonconforming FE-Discretisations with Application to Non-Matching Grids

Nonconforming finite element discretisations require special care in the construction of the prolongation and restriction in the multigrid process. In this paper, a general scheme is proposed, which guarantees the approximation property. As an example, the technique is applied to the discretisation by non-matching grids (mortar elements). Received: October 15, 1998     

Domain decomposition methods for the neutron diffusion problem

The neutronic simulation of a nuclear reactor core is performed using the neutron transport equation, and leads to an eigenvalue problem in the steady-state case. Among the deterministic resolution methods, simplified transport (S_PN$S{P}_N$) or diffusion approximations are often used. The MINOS solver developed at CEA Saclay uses a mixed dual finite element method for the resolution of these problems, and has shown his efficiency. In order to take into account the heterogeneities of the geometry, a very fine mesh is generally required, and leads to expensive calculations for industrial applications. In order to take advantage of parallel computers, and to reduce the computing time and the local memory requirement, we propose here two domain decomposition methods based on the MINOS solver. The first approach is a component mode synthesis method on overlapping subdomains: several eigenmodes solutions of a local problem on each subdomain are taken as basis functions used for the resolution of the global problem on the whole domain. The second approach is an iterative method based on a non-overlapping domain decomposition with Robin interface conditions. At each iteration, we solve the problem on each subdomain with the interface conditions given by the solutions on the adjacent subdomains estimated at the previous iteration. Numerical results on parallel computers are presented for the diffusion model on realistic 2D and 3D cores.     

The Streamline–Diffusion Method for Conforming and Nonconforming Finite Elements of Lowest Order Applied to Convection–Diffusion Problems

We consider the streamline-diffusion finite element method with finite elements of lowest order for solving convection-diffusion problems. Our investigations cover both conforming and nonconforming finite element approximations on triangular and quadrilateral meshes. Although the considered finite elements are of the same interpolation order their stability and approximation properties are quite different. We give a detailed overview on the stability and the convergence properties in the L ²- and in the streamline–diffusion norm. Numerical experiments show that often the theoretical predictions on the convergence properties are sharp. Received December 7, 1999; revised October 5, 2000     

Finite Element Methods for Eigenvalue Problems on a Rectangle with (Semi-) Periodic Boundary Conditions on a Pair of Adjacent Sides

This paper deals with a class of elliptic differential eigenvalue problems (EVPs) of second order on a rectangular domain Ω⊂ℝ², with periodic or semi-periodic boundary conditions (BCs) on two adjacent sides of Ω. On the remaining sides, classical Dirichlet or Robin type BCs are imposed. First, we pass to a proper variational formulation, which is shown to fit into the framework of abstract EVPs for strongly coercive, bounded and symmetric bilinear forms in Hilbert spaces. Next, the variational EVP serves as the starting point for finite element approximations. We consider finite element methods (FEMs) without and with numerical quadrature, both with triangular and with rectangular meshes. The aim of the paper is to show that well-known error estimates, established for finite element approximations of elliptic EVPs with classical BCs, remain valid for the present type of EVPs, including the case of multiple exact eigenvalues. Finally, the analysis is illustrated by a non-trivial numerical example, the exact eigenpairs of which can be determined. Received March 2, 1999; revised July 8, 1999     

A Solution Method for a Certain Nonlinear Interface Problem in Unbounded Domains

In this paper, we consider a kind of nonlinear interface problem in unbounded domains. To solve this problem, we discuss a new coupling of finite element and boundary element by adding an auxiliary circle. We first derive the optimal error estimate of finite element approximation to the coupled FEM-BEM problem. Then we introduce a preconditioning steepest descent method for solving the discrete system by constructing a cheap domain decomposition preconditioner. Moreover, we give a complete analysis to the convergence speed of this iterative method. Received March 30, 2000; revised November 29, 2000     

An Algebraic Multigrid Method for Higher-order Finite Element Discretizations

In this paper, we will design and analyze a class of new algebraic multigrid methods for algebraic systems arising from the discretization of second order elliptic boundary value problems by high-order finite element methods. For a given sparse stiffness matrix from a quadratic or cubic Lagrangian finite element discretization, an algebraic approach is carefully designed to recover the stiffness matrix associated with the linear finite element disretization on the same underlying (but nevertheless unknown to the user) finite element grid. With any given classical algebraic multigrid solver for linear finite element stiffness matrix, a corresponding algebraic multigrid method can then be designed for the quadratic or higher order finite element stiffness matrix by combining with a standard smoother for the original system. This method is designed under the assumption that the sparse matrix to be solved is associated with a specific higher order, quadratic for example, finite element discretization on a finite element grid but the geometric data for the underlying grid is unknown. The resulting new algebraic multigrid method is shown, by numerical experiments, to be much more efficient than the classical algebraic multigrid method which is directly applied to the high-order finite element matrix. Some theoretical analysis is also provided for the convergence of the new method.     

A 3D Crouzeix-Raviart mortar finite element

A mortar finite element discretization of the second order elliptic problem in three dimensions, on non-matching grids, using the 3D Crouzeix-Raviart (CR) finite element in each subdomain, is proposed in this paper. The overall discretization is based on using only the nodal values on the mortar side of a subdomain interface for the calculation of the mortar projection, as opposed to applying the conventional approach where some nodal values in the interior of a subdomain are also required. Since the interior degrees of freedom disappear completely from the computation of the mortar projection, the proposed algorithm becomes less intricate and more flexible as compared to the conventional approach. An error estimate is given, and some numerical experiments are presented.     

Coupling of Nonconforming Finite Elements and Boundary Elements I: A Priori Estimates

Nonconforming finite element methods are sometimes considered as a variational crime and so we may regard its coupling with boundary element methods. In this paper, the symmetric coupling of nonconforming finite elements and boundary elements is established and a priori error estimates are shown. The coupling involves a further continuous layer on the interface in order to separate the nonconformity in the domain from its boundary data which are required to be continuous. Numerical examples prove the new scheme useful in practice. A posteriori error control and adaptive algorithms will be studied in the forthcoming Part II. Received: November 26, 1997; revised February 10, 1999     

Numerical Integrations and Unit Resolution Multipliers for Domain Decomposition Methods with Nonmatching Grids

In this paper, we are concerned with the non-overlapping domain decomposition method (DDM) with nonmatching grids for three-dimensional problems. The weak continuity of the DDM solution on the interface is imposed by some Lagrange multiplier. We shall first analyze the influence of the numerical integrations over the interface on the (non-conforming) approximate solution. Then we will propose a simple approach to construct multiplier spaces, one of which can be simply spanned by some smooth basis functions with local compact supports, and thus makes the numerical integrations on the interface rather simple and inexpensive. Also it is shown this multiplier space can generate an optimal approximate solution. Numerical results are presented to compare the new method with the point to point method widely used in engineering.     

The Fourier Finite-element Approximation of the Lamé Equations in Axisymmetric Domains with Edges

This paper is concerned with a priori error estimates and convergence analysis of the Fourier-finite-element solutions of the Neumann problem for the Lamé equations in axisymmetric domains with reentrant edges. The Fourier-FEM combines the approximating Fourier method with respect to the rotational angle using trigonometric polynomials of degree N (N→∞), with the finite element method on the plane meridian domain of with mesh size h (h→0) for approximating the Fourier coefficients. The asymptotic behavior of the solution near reentrant edges is described by singularity functions in non-tensor product form and treated numerically by means of finite element method on locally graded meshes. For the rate of convergence of the combined approximations in is proved to be of the order      

Asymptotic boundary conditions with immersed finite elements for interface magnetostatic/electrostatic field problems with open boundary

Many of the magnetostatic/electrostatic field problems encountered in aerospace engineering, such as plasma sheath simulation and ion neutralization process in space, are not confined to finite domain and non-interface problems, but characterized as open boundary and interface problems. Asymptotic boundary conditions (ABC) and immersed finite elements (IFE) are relatively new tools to handle open boundaries and interface problems respectively. Compared with the traditional truncation approach, asymptotic boundary conditions need a much smaller domain to achieve the same accuracy. When regular finite element methods are applied to an interface problem, it is necessary to use a body-fitting mesh in order to obtain the optimal convergence rate. However, immersed finite elements possess the same optimal convergence rate on a Cartesian mesh, which is critical to many applications. This paper applies immersed finite element methods and asymptotic boundary conditions to solve an interface problem arising from electric field simulation in composite materials with open boundary. Numerical examples are provided to demonstrate the high global accuracy of the IFE method with ABC based on Cartesian meshes, especially around both interface and boundary. This algorithm uses a much smaller domain than the truncation approach in order to achieve the same accuracy.     

Interface element method: Treatment of non-matching nodes at the ends of interfaces between partitioned domains

A methodology for interface element method (IEM) to combine partitioned domains with non-matching nodes at the ends of interfaces is presented. The IEM is introduced to satisfy the continuity and the compatibility conditions on non-matching interfaces between partitioned finite element domains. Interface elements are defined on the finite elements bordering on the interfaces, and the moving least square (MLS) approximations are employed to construct the shape functions of the interface elements. By modifying the shape functions of the interface elements at the ends of non-matching interfaces, partitioned domains are glued such that all properties of the IEM are satisfied. The modifications are made to sub-domains and weight functions in the MLS approximations. The numerical examples show that the present IEM is very effective for the analysis of a partitioned system and for global-local analysis.     

The Inf-Sup Condition for the Mapped Q k ?P k?1 disc Element in Arbitrary Space Dimensions

 One of the most popular pairs of finite elements for solving mixed formulations of the Stokes and Navier–Stokes problem is the Q _k −P _k−1 ^disc element. Two possible versions of the discontinuous pressure space can be considered: one can either use an unmapped version of the P _k−1 ^disc space consisting of piecewise polynomial functions of degree at most k−1 on each cell or define a mapped version where the pressure space is defined as the image of a polynomial space on a reference cell. Since the reference transformation is in general not affine but multilinear, the two variants are not equal on arbitrary meshes. It is well-known, that the inf-sup condition is satisfied for the first variant. In the present paper we show that the latter approach satisfies the inf-sup condition as well for k≥2 in any space dimension. Received January 31, 2001; revised May 2, 2002 Published online: July 26, 2002     

<Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-interpolation on quadrilateral and hexahedral meshes with hanging nodes

We propose a Scott-Zhang type finite element interpolation operator of first order for the approximation of H ¹-functions by means of continuous piecewise mapped bilinear or trilinear polynomials. The novelty of the proposed interpolation operator is that it is defined for general non-affine equivalent quadrilateral and hexahedral elements and so-called 1-irregular meshes with hanging nodes. We prove optimal local approximation properties of this interpolation operator for functions in H ¹. As necessary ingredients we provide a definition of a hanging node and a rigorous analysis of the issue of constrained approximation which cover both the two- and three-dimensional case in a unified fashion.