Higher order unfitted FEM for Stokes interface problems

We consider the discretization of a stationary Stokes interface problem in a velocity-pressure formulation. The interface is described implicitly as the zero level of a scalar function as it is common in level set based methods. Hence, the interface is not aligned with the mesh. An unfitted finite element discretization based on a Taylor-Hood velocity-pressure pair and an XFEM (or CutFEM) modification is used for the approximation of the solution. This allows for the accurate approximation of solutions which have strong or weak discontinuities across interfaces which are not aligned with the mesh. To arrive at a consistent, stable and accurate formulation we require several additional techniques. First, a Nitsche-type formulation is used to implement interface conditions in a weak sense. Secondly, we use the ghost penalty stabilization to obtain an inf-sup stable variational formulation. Finally, for the highly accurate approximation of the implicitly described geometry, we use a combination of a piecewise linear interface reconstruction and a parametric mapping of the underlying mesh. We introduce the method and discuss results of numerical examples. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

A new alternating positive semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models

Based on the special positive semidefinite splittings of the saddle point matrix, we propose a new alternating positive semidefinite splitting (APSS) iteration method for the saddle point problem arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current problem. We prove that the new APSS iteration method is unconditionally convergent for both cases of the simple topology and the general topology. The new APSS matrix can be used as a preconditioner to accelerate the convergence rate of Krylov subspace methods. Numerical results show that the new APSS preconditioner is superior to the existing preconditioners.     

On the convergence of finite element solutions to the interface problem for the Stokes system

The Stokes system with a discontinuous coefficient (Stokes interface problem) and its finite element approximations are considered. We firstly show a general error estimate. To derive explicit convergence rates, we introduce some appropriate assumptions on the regularity of exact solutions and on a geometric condition for the triangulation. We mainly deal with the MINI element approximation and then consider P1-iso-P2/P1 element approximation. Results are expected to give an instructive remark in numerical analysis for two-phase flow problems.     

基于非协调元的区域分解法──强重迭情形

基于非协调元的区域分解法──强重迭情形黄建国（上海交通大学应用数学系）ＡＤＯＭＡＩＮＤＥＣＯＭＰＯＳＩＴＩＯＮＭＥＴＨＯＤＦＯＲＮＯＮＣＯＮＦＯＲＭＩＮＧＦＩＮＩＴＥＥＬＥＭＥＮＴ──ＴＨＥＣＡＳＥＯＦＳＴＲＯＮＧＯＶＥＲＬＡＰ￥ＨｕａｎｇＪｉａｎ...     

Analysis of the Fictitious Domain Method with an L 2-Penalty for Elliptic Problems

The L ²-penalty fictitious domain method is based on a reformulation of the original problem in a larger simple-shaped domain by introducing a discontinuous reaction term with a penalty parameter ε > 0. We first derive regularity results and some a priori estimates and then prove several error estimates. We also give several error estimates for discretization problems by the finite element and finite volume methods.     

Numerical analysis for a locally damped wave equation

We consider a semi-discrete finite element formulation with artificial viscosity for the numerical approximation of a problem that models the damped vibrations of a string with fixed ends. The damping coefficient depends on the spatial variable and is effective only in a sub-interval of the domain. For this scheme, the energy of semi-discrete solutions decays exponentially and uniformly with respect to the mesh parameter to zero. We also introduce an implicit in time discretization. Error estimates for the semi-discrete and fully discrete schemes in the energy norm are provided and numerical experiments performed.     

Numerical investigation of a model problem for the Poisson equation with inequality constraints in a domain with a cut

A model problem is considered for the Poisson equation in a two-dimensional domain with a cut. The Dirichlet and Neumann conditions are imposed on the exterior boundary of the domain together with the nonnegativity condition for the jump across the edges of the cut. In addition, the absolute value of the gradient inside the domain must be bounded by some constant. The boundary value problem turns into a variational problem, and the unknown function must yield the minimum of the energy functional on some convex set. After discretization of the problem by the finite element method, an Uzawa-type algorithm is used to find a solution. Some examples are included of solving the discrete problem.     

Nitsche’s Method for a Transport Problem in Two-phase Incompressible Flows

We consider a parabolic interface problem which models the transport of a dissolved species in two-phase incompressible flow problems. Due to the so-called Henry interface condition the solution is discontinuous across the interface. We use an extended finite element space combined with a method due to Nitsche for the spatial discretization of this problem and derive optimal discretization error bounds for this method. For the time discretization a standard θ-scheme is applied. Results of numerical experiments are given that illustrate the convergence properties of this discretization.     

Preconditioned iterative methods for diffusion problems with high‐contrast inclusions

This paper is concerned with robust numerical treatment of an elliptic PDE with high‐contrast coefficients, for which classical finite‐element discretizations yield ill‐conditioned linear systems. This paper introduces a procedure by which the discrete system obtained from a linear finite element discretization of the given continuum problem is converted into an equivalent linear system of the saddle‐point type. Three preconditioned iterative procedures—preconditioned Uzawa, preconditioned Lanczos, and preconditioned conjugate gradient for the square of the matrix—are discussed for a special type of the application, namely, highly conducting particles distributed in the domain. Robust preconditioners for solving the derived saddle‐point problem are proposed and investigated. Robustness with respect to the contrast parameter and the discretization scale is also justified. Numerical examples support theoretical results and demonstrate independence of the number of iterations of the proposed iterative schemes on the contrast in parameters of the problem and the mesh size.     

On Preconditioning of Incompressible Non-Newtonian Flow Problems

This paper deals with fast and reliable numerical solution methods for the incompressible non-Newtonian Navier-Stokes equations. To handle the nonlinearity of the governing equations, the Picard and Newton methods are used to linearize these coupled partial differential equations. For space discretization we use the finite element method and utilize the two-by-two block structure of the matrices in the arising algebraic systems of equations. The Krylov subspace iterative methods are chosen to solve the linearized discrete systems and the development of computationally and numerically efficient preconditioners for the two-by-two block matrices is the main concern in this paper. In non-Newtonian flows, the viscosity is not constant and its variation is an important factor that affects the performance of some already known preconditioning techniques. In this paper we examine the performance of several preconditioners for variable viscosity applications, and improve them further to be robust with respect to variations in viscosity.     

Continuous preconditioners for the mixed Poisson problem

In this note, we show how to apply preconditioners designed for piecewise linear finite element discretizations of the Poisson problem as preconditioners for the mixed problem. Our preconditioner can be applied both to the original and to the reduced Schur complement problem. Combined with a suitable iterative method, the number of iterations required to solve the preconditioned system will have the same dependency on the mesh size as for the preconditioner applied to the Poisson problem. The presented theory is demonstrated by numerical examples.     

基于二重网格的定常Navier-Stokes方程的局部和并行有限元算法

对二维定常的不可压缩的Navier-Stokes方程的局部和并行算法进行了研究．给出的算法是多重网格和区域分解相结合的算法,它是基于两个有限元空间:粗网格上的函数空间和子区域的细网格上的函数空间．局部算法是在粗网格上求一个非线性问题,然后在细网格上求一个线性问题,并舍掉内部边界附近的误差相对较大的解．最后,基于局部算法,通过有重叠的区域分解而构造了并行算法,并且做了算法的误差分析,得到了比标准有限元方法更好的误差估计,也对算法做了数值试验,数值结果通过比较验证了本算法的高效性和合理性．     

Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition

In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases.     

Unfitted finite element methods for the heat conduction in composite media with contact resistance

This article is concerned with the heat conduction problem in composite media. In practical applications, the composite materials often do not contact well and there exist gaps between the contacting materials. This leads to the thermal contact resistance effect which results in a discontinuity of the temperature across the interface. In this article, an unfitted finite element method is proposed to solve the problem. Different from the traditional finite element method, the proposed method uses structured meshes that allow the interface to cut through. To avoid integrating on curved domains and interfaces, the interface is approximated by a broken line/plane corresponding to the triangulation. In addition, a ghost‐penalty is added to recover the condition number of the stiffness matrix to with a hidden constant independent of the mesh‐interface geometry. A rigorous analysis is provided. Finally, numerical tests are presented to verify the theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 354–380, 2017     

HYBRID ALGEBRAIC MULTILEVEL PRECONDITIONING METHODS

ＨＹＢＲＩＤＡＬＧＥＢＲＡＩＣＭＵＬＴＩＬＥＶＥＬＰＲＥＣＯＮＤＩＴＩＯＮＩＮＧＭＥＴＨＯＤＳ￥ＢａｉＺｈｏｎｇｚｈｉ（白中治）（ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，复旦大学，邮编：２００４３３）Ａｂｓｔｒａｃｔ：Ａｃｌａｓｓｏｆｈｙｂｒｉｄａｌｇｅｂｒ...     

Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Efficient multilevel preconditioners are developed and analyzed for the quadrature finite element Galerkin approximation of the biharmonic Dirichlet problem. The quadrature scheme is formulated using the Bogner–Fox–Schmit rectangular element and the product two‐point Gaussian quadrature. The proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the quadrature scheme. The preconditioners are implemented by optimal algorithms, and they are used to accelerate convergence of the preconditioned conjugate gradient method. Numerical results are presented demonstrating efficiency of the preconditioners. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data

This article concerns numerical approximation of a parabolic interface problem with general $L^2$ initial value. The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface, with piecewise linear approximation to the interface. The semi-discrete finite element problem is furthermore discretized in time by the $k$-step backward difference formula with $ k=1,\ldots,6 $. To maintain high-order convergence in time for possibly nonsmooth $L^2$ initial value, we modify the standard backward difference formula at the first $k-1$ time levels by using a method recently developed for fractional evolution equations. An error bound of $\mathcal{O}(t_n^{-k}\tau^k+t_n^{-1}h^2|\log h|)$ is established for the fully discrete finite element method for general $L^2$ initial data.     

How to Recover the Convergent Rate for Richardson Extrapolation on Bounded Domains

We are interested in solving elliptic problems on bounded convex domains by higher order methods using the Richardson extrapolation. The theoretical basis for the application of the Richardson extrapolation is the asymptotic error expansion with a remainder of higher order. Such an expansion has been derived by the method of finite difference, where, in the neighborhood of the boundary one must reject the elementary difference analogs and adopt complex ones. This plight can be changed if we turn to the method of finite elements, where no additional boundary approximation is needed but an easy triangulation is chosen, i.e. the higher order boundary approximation is replaced by a chosen triangulation. Specifically, a global error expansion with a remainder of fourth order can be derived by the linear finite element discretization over a chosen triangulation, which is obtained by decomposing the domain first and then subdividing each subdomain almost uniformly. A fourth order method can thus be constructed by the simplest linear finite element approximation over the chosen triangulation using the Richardson extrapolation.     

Nitsche-XFEM for a transport problem in two-phase incompressible flows

We consider the transport of a dissolved species in a divergence-free immiscible incompressible two-phase flow modeled by a convection diffusion equation. The so-called Henry interface condition leads to a jump condition for the concentration at the interface between the two phases. This discontinuity of the solution render the numerical solution on unfitted meshes difficult. Furthermore time discretization on moving interfaces and handling typically convection dominant situations makes the overall problem delicate. We propose a numerical method using extended finite elements and a Nitsche-type technique combined with streamline diffusion stabilization. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)