An iterative adaptive finite element method for elliptic eigenvalue problems

We consider the task of resolving accurately the n$n$th eigenpair of a generalized eigenproblem rooted in some elliptic partial differential equation (PDE), using an adaptive finite element method (FEM). Conventional adaptive FEM algorithms call a generalized eigensolver after each mesh refinement step. This is not practical in our situation since the generalized eigensolver needs to calculate n$n$ eigenpairs after each mesh refinement step, it can switch the order of eigenpairs, and for repeated eigenvalues it can return an arbitrary linear combination of eigenfunctions from the corresponding eigenspace. In order to circumvent these problems, we propose a novel adaptive algorithm that only calls a generalized eigensolver once at the beginning of the computation, and then employs an iterative method to pursue a selected eigenvalue–eigenfunction pair on a sequence of locally refined meshes. Both Picard’s and Newton’s variants of the iterative method are presented. The underlying partial differential equation (PDE) is discretized with higher-order finite elements (hp$hp$-FEM) but the algorithm also works for standard low-order FEM. The method is described and accompanied with theoretical analysis and numerical examples. Instructions on how to reproduce the results are provided.     

Higher order weakly over-penalized symmetric interior penalty methods

In this paper we study higher order weakly over-penalized symmetric interior penalty methods for second-order elliptic boundary value problems in two dimensions. We derive h$h$–p$p$ error estimates in both the energy norm and the _L2$L_2$ norm and present numerical results that corroborate the theoretical results.     

A remark on least-squares mixed element methods for reaction–diffusion problems

In this paper, we propose a least-squares mixed element procedure for a reaction–diffusion problem based on the first-order system. By selecting the least-squares functional properly, the resulting procedure can be split into two independent symmetric positive definite schemes, one of which is for the unknown variable and the other of which is for the unknown flux variable, which lead to the optimal order H¹(Ω)$H^1(\Omega )$ and L²(Ω)$L^2(\Omega )$ norm error estimates for the primal unknown and optimal H(div;Ω)$H(\mathrm{div};\Omega )$ norm error estimate for the unknown flux. Finally, we give some numerical examples.     

New expansions of numerical eigenvalues by Wilson’s element

The paper explores new expansions of eigenvalues for −Δu=λρu$-\Delta u=\lambda \rho u$ in S$S$ with Dirichlet boundary conditions by Wilson’s element. The expansions indicate that Wilson’s element provides lower bounds of the eigenvalues. By the extrapolation or the splitting extrapolation, the O(h⁴)$O\left(h^4\right)$ convergence rate can be obtained, where h$h$ is the maximal boundary length of uniform rectangles. Numerical experiments are carried to verify the theoretical analysis made. It is worth pointing out that these results are new, compared with the recent book, Lin and Lin [Q. Lin, J. Lin, Finite Element Methods; Accuracy and Improvement, Science Press, Beijing, 2006].     

Equidistributed error mesh for problems with exponential boundary layers

In this paper we present a new piecewise-linear finite element mesh suitable for the discretization of the one-dimensional convection–diffusion equation -εu^″-bu^′=0$-\epsilon u^{″}-{\mathit{bu}}^{\prime }=0$, u(0)=0$u(0)=0$, u(1)=1$u(1)=1$. The solution to this equation exhibits an exponential boundary layer which occurs also in more complicated convection–diffusion problems of the form -εΔu-b∂u/∂x+cu=f$-\epsilon \mathrm{\Delta }u-b\partial u/\partial x+\mathit{cu}=f$. The new mesh is based on the equidistribution of the interpolation error and it takes into account finite computer arithmetic. It is demonstrated numerically that for the above problem, the new mesh has remarkably better convergence properties than the well-known Shishkin and Bakhvalov meshes.     

Linear and bilinear immersed finite elements for planar elasticity interface problems

This article is to discuss the linear (which was proposed in  and ) 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 L²$L^2$ and semi-H¹$H^1$ norms.     

A comparison of a posteriori error estimates for biharmonic problems solved by the FEM

The classical a posteriori error estimates are mostly oriented to the use in the finite element h$h$-methods while the contemporary higher-order hp$hp$-methods usually require new approaches in a posteriori error estimation. These methods hold a very important position among adaptive numerical procedures for solving ordinary as well as partial differential equations arising from various technical applications.     

Error estimation of a quadratic finite volume method on right quadrangular prism grids

In this paper, we develop a finite volume element method with affine quadratic bases on right quadrangular prism meshes for three-dimensional elliptic boundary value problems. The optimal H¹$H^1$-norm error estimate of second order accuracy is proved under certain assumptions about the meshes. Numerical results are presented to illustrate the theoretical analysis.     

Two-grid methods for finite volume element approximations of nonlinear parabolic equations

Two-grid methods are studied for solving a two dimensional nonlinear parabolic equation using finite volume element method. The methods are based on one coarse-grid space and one fine-grid space. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine-grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h=O(H³|lnH|)$h=O\left(H^3|lnH|\right)$. As a result, solving such a large class of nonlinear parabolic equations will not be much more difficult than solving one single linearized equation.     

Analysis of one-dimensional Helmholtz equation with PML boundary

In this paper, the linear conforming finite element method for the one-dimensional Bérenger's PML boundary is investigated and well-posedness of the given equation is discussed. Furthermore, optimal error estimates and stability in the L²$L^2$ or H¹$H^1$-norm are derived under the assumption that h$h$, h²ω²$h^2{\omega }^2$ and h²ω³$h^2{\omega }^3$ are sufficiently small, where h$h$ is the mesh size and ω$\omega$ denotes a fixed frequency. Numerical examples are presented to validate the theoretical error bounds.     

Time discretization and quantization methods for optimal multiple switching problem

In this paper, we study probabilistic numerical methods based on optimal quantization algorithms for computing the solution to optimal multiple switching problems with regime-dependent state process. We first consider a discrete-time approximation of the optimal switching problem, and analyse its rate of convergence. Given a time step h$h$, the error is in general of order (hlog(1/h))^1/2${(hlog(1/h))}^{1/2}$, and of order h^1/2$h^{1/2}$ when the switching costs do not depend on the state process. We next propose quantization numerical schemes for the space discretization of the discrete-time Euler state process. A Markovian quantization approach relying on the optimal quantization of the normal distribution arising in the Euler scheme is analysed. In the particular case of uncontrolled state process, we describe an alternative marginal quantization method, which extends the recursive algorithm for optimal stopping problems as in Bally (2003) [1]. A priori L^p$L^p$-error estimates are stated in terms of quantization errors. Finally, some numerical tests are performed for an optimal switching problem with two regimes.     

Convergence of block iterative methods for linear systems with generalized H-matrices

The paper studies the convergence of some block iterative methods for the solution of linear systems when the coefficient matrices are generalized H$H$-matrices. A truth is found that the class of conjugate generalized H$H$-matrices is a subclass of the class of generalized H$H$-matrices and the convergence results of R. Nabben [R. Nabben, On a class of matrices which arises in the numerical solution of Euler equations, Numer. Math. 63 (1992) 411–431] are then extended to the class of generalized H$H$-matrices. Furthermore, the convergence of the block AOR iterative method for linear systems with generalized H$H$-matrices is established and some properties of special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper.     

An inexact implicit method for general mixed variational inequalities

In the present paper, we present an inexact implicit method with a variable parameter for general mixed variational inequalities. We use a self-adaptive technique to adjust parameter ρ$\rho$ at each iteration. The main advantage of this technique is that the method can adjust the parameter automatically and the numbers of iteration are not very sensitive to different initial parameter _ρ0.${\rho }_0.$