Additive Schwarz method for the p-version of the boundary element method for the single layer potential operator on a plane screen

Summary. We analyze an additive Schwarz preconditioner for the p-version of the boundary element method for the single layer potential operator on a plane screen in the three-dimensional Euclidean space. We decompose the ansatz space, which consists of piecewise polynomials of degree p on a mesh of size h, by introducing a coarse mesh of size . After subtraction of the coarse subspace of piecewise constant functions on the coarse mesh this results in local subspaces of piecewise polynomials living only on elements of size H. This decomposition yields a preconditioner which bounds the spectral condition number of the stiffness matrix by . Numerical results supporting the theory are presented. Received August 15, 1998 / Revised version received November 11, 1999 / Published online December 19, 2000     

Optimal control of the Stokes equations: conforming and non-conforming finite element methods under reduced regularity

This paper deals with a control-constrained linear-quadratic optimal control problem governed by the Stokes equations. It is concerned with situations where the gradient of the velocity field is not bounded. The control is discretized by piecewise constant functions. The state and the adjoint state are discretized by finite element schemes that are not necessarily conforming. The approximate control is constructed as projection of the discrete adjoint velocity in the set of admissible controls. It is proved that under certain assumptions on the discretization of state and adjoint state this approximation is of order 2 in L ²(Ω). As first example a prismatic domain with a reentrant edge is considered where the impact of the edge singularity is counteracted by anisotropic mesh grading and where the state and the adjoint state are approximated in the lower order Crouzeix-Raviart finite element space. The second example concerns a nonconvex, plane domain, where the corner singularity is treated by isotropic mesh grading and state and adjoint state can be approximated by a couple of standard element pairs.     

Analysis of a finite element method for pressure/potential formulation of elastoacoustic spectral problems

A finite element method to approximate the vibration modes of a structure enclosing an acoustic fluid is analyzed. The fluid is described by using simultaneously pressure and displacement potential variables, whereas displacement variables are used for the solid. A mathematical analysis of the continuous spectral problem is given. The problem is discretized on a simplicial mesh by using piecewise constant elements for the pressure and continuous piecewise linear finite elements for the other fields. Error estimates are settled for approximate eigenvalues and eigenfrequencies. Finally, implementation issues are discussed.

     

Fourier–Finite-element Approximation of Elliptic Interface Problems in Axisymmetric Domains

The paper deals with the Fourier-finite-element method (FFEM), which combines the approximate Fourier method with the finite-element method, and its application to Poisson-like equations −p̂Δ₃û = f̂ in three-dimensional axisymmetric domains Ωˆ. Here, pˆ is a piecewise constant coefficient having a jump at some axisymmetric interface. Special emphasis is given to estimates of the Fourier-finite-element error in the Sobolev space H¹(Ωˆ), if the interface is smooth or if it meets the boundary of Ωˆ at some edge. In general, the solution û contains a singularity at the interface, which is described by a tensor product representation and treated numerically by appropriate mesh grading in the meridian plane of Ωˆ. The rate of convergence of the combined approximation in H¹(Ωˆ) is proved to be 𝒪(h+N⁻¹) (h, N: the parameters of the finite-element- and Fourier-approximation, with h→0, N→∞). The theoretical results are confirmed by numerical experiments.     

Analysis of an algorithm for generating locally optimal meshes for approximation by discontinuous piecewise polynomials

This paper discusses the problem of constructing a locally optimal mesh for the best approximation of a given function by discontinuous piecewise polynomials. In the one-dimensional case, it is shown that, under certain assumptions on the approximated function, Baines' algorithm [M.J. Baines, Math. Comp., 62 (1994), pp. 645-669] for piecewise linear or piecewise constant polynomials produces a mesh sequence which converges to an optimal mesh. The rate of convergence is investigated. A two-dimensional modification of this algorithm is proposed in which both the nodes and the connection between the nodes are self-adjusting. Numerical results in one and two dimensions are presented.

     

A uniformly convergent B-spline collocation method on a nonuniform mesh for singularly perturbed one-dimensional time-dependent linear convection–diffusion problem

A numerical method is proposed for solving singularly perturbed one-dimensional parabolic convection–diffusion problems. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on a uniform mesh by means of Rothe's method and B-spline collocation method in spatial direction on a piecewise uniform mesh of Shishkin type. The method is shown to be unconditionally stable and accurate of order O((Δx)²+Δt). An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. Several numerical experiments have been carried out in support of the theoretical results. Comparisons of the numerical solutions are performed with an upwind finite difference scheme on a piecewise uniform mesh and exponentially fitted method on a uniform mesh to demonstrate the efficiency of the method.     

A finite volume method on general surfaces and its error estimates

In this paper, we study a finite volume method and its error estimates for the numerical solution of some model second order elliptic partial differential equations defined on a smooth surface. The discretization is defined via a surface mesh consisting of piecewise planar triangles and piecewise polygons. The optimal error estimates of the approximate solution are proved in both the H¹ and L² norms which are of first order and second order respectively under mesh regularity assumptions. Some numerical tests are also carried out to experimentally verify our theoretical analysis.     

Extrapolation for the Approximations to the Solution of a Boundary Integral Equation on Polygonal Domains

In this paper, we consider a boundary integral equation of second kind rising from potential theory. The equation may be solved numerically by Galerkin's method using piecewise constant functions. Because of the singularities produced by the corners, we have to grade the mesh near the corner. In general, Chandler obtained the order 2 superconvergence of the iterated Galerkin solution in the uniform norm. It is proved in this paper that the Richardson extrapolation increases the accuracy from order 2 to order 4.     

Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation

In this paper, a piecewise constant time-stepping discontinuous Galerkin method combined with a piecewise linear finite element method is applied to solve control constrained optimal control problem governed by time fractional diffusion equation. The control variable is approximated by variational discretization approach. The discrete first-order optimality condition is derived based on the first discretize then optimize approach. We demonstrate the commutativity of discretization and optimization for the time-stepping discontinuous Galerkin discretization. Since the state variable and the adjoint state variable in general have weak singularity near t =?0and t = T, a time adaptive algorithm is developed based on step doubling technique, which can be used to guide the time mesh refinement. Numerical examples are given to illustrate the theoretical findings.     

Three-Dimensional Delaunay Mesh Generation

We propose an algorithm to compute a conforming Delaunay mesh of a bounded domain in specified by a piecewise linear complex. Arbitrarily small input angles are allowed, and the input complex is not required to be a manifold. Our algorithm encloses the input edges with a small buffer zone, a union of balls whose sizes are proportional to the local feature sizes at their centers. In the output mesh, the radius-edge ratio of the tetrahedra outside the buffer zone is bounded by a constant independent of the domain, while that of the tetrahedra inside the buffer zone is bounded by a constant depending on the smallest input angle. Furthermore, the output mesh is graded. Our work is the first that provides quality guarantees for Delaunay meshes in the presence of small input angles.     

Ultraconvergence of the derivative of high-order finite element method for elliptic problems with constant coefficients

In this article, for second order elliptic problems with constant coefficients, the local ultraconvergence of the derivative of finite element method using piecewise polynomials of degrees k (k ≥ 2) is studied by the interpolation postprocessing technique. Under suitable regularity and mesh conditions, we prove that at an interior vertex, which is away from the boundary with a fixed distance, the gradient of the postprecessed finite element solution using piecewise polynomials of degrees k (k ≥ 2) converges to the gradient of the exact solution with order . Numerical experiments are used to illustrate our theoretical findings.     

B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.     

Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: ε-uniformly convergent schemes

The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted grids is based on a majorant of the singular component of the grid solution that makes it possible to a priori find a subdomain in which the grid solution should be further refined given the perturbation parameter ε, the size of the uniform mesh in x, the desired accuracy of the grid solution, and the prescribed number of iterations K used to refine the solution. In the subdomains where the solution is refined, the grid problems are solved on uniform grids. The error of the solution thus constructed weakly depends on ε. The scheme converges almost ε-uniformly; namely, it converges under the condition N ^?1 = o(ε^v), where v = v(K) can be chosen arbitrarily small when K is sufficiently large. If a piecewise uniform grid is used instead of a uniform one at the final Kth iteration, the difference scheme converges ε-uniformly. For this piecewise uniform grid, the ratio of the mesh sizes in x on the parts of the mesh with a constant size (outside the boundary layer and inside it) is considerably less than that for the known ε-uniformly convergent schemes on piecewise uniform grids.     

High-Order Compact Finite-Difference Scheme for Singularly-Perturbed Reaction-Diffusion Problems on a New Mesh of Shishkin Type

In this work, a high-order compact finite-difference (HOCFD) scheme has been proposed to solve 1-dimensional (1D) and 2-dimensional (2D) elliptic and parabolic singularly-perturbed reaction-diffusion problems. A new kind of piecewise uniform mesh of Shishkin type (Miller et al. in Fitted Numerical Methods for Singular Perturbation Problems, 1996) has also been proposed and using this mesh the HOCFD scheme gives better results as compared to the results using the Shishkin mesh. Moreover, the stated method gives ε-uniform convergence and improved orders of convergence which have also been provided in the results for some test problems.     

On a generalized $L_2$ projection and some related stability estimates in Sobolev spaces

Summary. In this paper we investigate a stability estimate needed in hybrid finite and boundary element methods, especially in hybrid coupled domain decomposition methods including mortar finite elements. This stability estimate is equivalent to the stability of a generalized projection in certain Sobolev spaces. Using piecewise linear trial spaces and appropriate piecewise constant test spaces, the stability of the generalized projection is proved assuming some mesh conditions locally. Received April 11, 2000 / Revised version received February 15, 2001 / Published online July 25, 2001     

A note on a lowest order divergence‐free Stokes element on quadrilaterals

This short note reports a lowest order divergence‐free Stokes element on quadrilateral meshes. The velocity space is based on a P₁ spline element over the crisscross partition of a quadrilateral, and the pressure is approximated by piecewise constant. For a given quadrilateral mesh, this element is stable if and only if the well‐known Q₁‐P₀ element is also stable. Although this method is a subspace method of Qin‐Zhang's P₁‐P₀ element, their velocity solutions are precisely equal. Moreover, an explicit basis representation is also provided. These theoretical findings are verified by numerical tests.     

热传导型半导体器件的二次元特征有限体积元方法及分析:H1模估计

该文构造了热传导型半导体器件的全离散特征有限体积元格式,将特征线方法与有限体积元方法相结合, 采用Lagrange型分片二次多项式空间和分片常数函数空间分别作为试探函数和检验函数空间,并进行误差分析,得到了最优阶H¹模误差估计结果.     

An iterative substructuring method for the p-version of the boundary element method for hypersingular integral operators in three dimensions

Summary. We study preconditioners for the -version of the boundary element method for hypersingular integral equations in three dimensions. The preconditioners are based on iterative substructuring of the underlying ansatz spaces which are constructed by using discretely harmonic basis functions. We consider a so-called wire basket preconditioner and a non-overlapping additive Schwarz method based on the complete natural splitting, i.e. with respect to the nodal, edge and interior functions, as well as an almost diagonal preconditioner. In any case we add the space of piecewise bilinear functions which eliminate the dependence of the condition numbers on the mesh size. For all these methods we prove that the resulting condition numbers are bounded by . Here, is the polynomial degree of the ansatz functions and is a constant which is independent of and the mesh size of the underlying boundary element mesh. Numerical experiments supporting these results are reported. Received July 8, 1996 / Revised version received January 8, 1997     

A streamline diffusion finite element method on a shishkin mesh for a convection-diffusion problem

We consider a model time-dependent convection-diffusion problem, whose solution may exhibit interior and boundary layers. The standard streamline diffusion scheme, with piecewise linear elements on a uniform mesh, will converge only at points that are not close to any layer. We replace the uniform mesh by a special piecewise uniform mesh that is chosen a priori and resolves part of any outflow boundary layer. The resulting method is convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 away from layers and almost order 3/4 inside the boundary layer.     

Fourth order algorithms for a semilinear singular perturbation problem

Fourth order finite-difference algorithms for a semilinear singularly perturbed reaction–diffusion problem are discussed and compared both theoretically and numerically. One of them is the method of Sun and Stynes (1995) which uses a piecewise equidistant discretization mesh of Shishkin type. Another one is a simplified version of Vulanović's method (1993), based on a discretization mesh of Bakhvalov type. It is shown that the Bakhvalov mesh produces much better numerical results. This revised version was published online in June 2006 with corrections to the Cover Date.