Zi-Cai Li

Boundary methods for solving elliptic problems with singularities and interfaces

Boundary approximation techniques are described for solving homogeneous self-adjoint elliptic equations. Piecewise expansions into particular solutions are used which approximate both the boundary and interface conditions in a least squares sense. Convergence of such approximations is proved and error estimates are derived in a natural norm. Numerical experiments are reported for the singular Motz problem which yield extremely accurate solutions with only a modest computational effort.

Combined methods for elliptic equations with singularities, interfaces, and infinities

Preface. Introduction. Part I: Singularities, Treatments and Combinations. 1. Different Numerical Methods. 2. Singularities and Treatments. Part II: Combined Methods. 3. Boundary Approximation Methods. 4. Combinations of RGM and FEM. 5. Combinations of Various FEMs. Part III: Coupling Techniques. 6. Lagrange Multipliers and Other Coupling Techniques. 7. Penalty Techniques. 8. Simplified Hybrid Methods. 9. Penalty Plus Hybrid Techniques. 10. Optimal Combinations for Various FEMs. 11. Combinations of RGM and FDM. Part IV: Applications and Advances Topics. 12. Crack-Infinity Problem. 13. Wind Flow Over Buildings. 14. Global Superconvergence in Combinations. 15. Iterative Substructing Methods. 16. Schwarz Alternating Method. Epilogue. Reference. Glossary of Symbols. Index.

A regional decomposition method for recognizing handprinted characters

A regional decomposition method is proposed to facilitate pattern analysis and recognition. It splits a complicated pattern into several simple parts or sub-patterns, so that the pattern can be identified by examining the distinct parts. A complexity analysis is derived in this paper to prove the effectiveness of the regional decomposition method; mathematical and statistical formulas are also provided to evaluate the recognition rates of different parts. For a sample of 36 alphanumeric characters handprinted in 89 most common styles, the total mean recognition rates of parts have been found to be 30% higher than those obtained from subjective experiments. >

New expansions of numerical eigenvalues for -Δ= by nonconforming elements

The paper explores new expansions of the eigenvalues for -Δu = Λpu in S with Dirichlet boundary conditions by the bilinear element (denoted Q 1 ) and three nonconforming elements, the rotated bilinear element (denoted Q 1 rot ), the extension of Q 1 rot (denoted EQ 1 rot ) and Wilsons elements. The expansions indicate that Q 1 and Q 1 rot provide upper bounds of the eigenvalues, and that EQ 1 rot and Wilsons elements provide lower bounds of the eigenvalues. By extrapolation, the O(h 4 ) convergence rate can be obtained, where h is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.

Boundary penalty finite element methods for blending surfaces, III. Superconvergence and stability and examples

Abstract This paper is Part III of the study on blending surfaces by partial differential equations (PDEs). The blending surfaces in three dimensions (3D) are taken into account by three parametric functions, x ( r , t ), y ( r , t ) and z ( r , t ). The boundary penalty techniques are well suited to the complicated tangent (i.e., normal derivative) boundary conditions in engineering blending. By following the previous papers, Parts I and II in Li (J. Comput. Math. 16 (1998) 457–480; J. Comput. Appl. Math. 110 (1999) 155–176) the corresponding theoretical analysis is made to discover that when the penalty power σ =2, σ =3 ( or 3.5) and 0 σ ⩽1.5 in the boundary penalty finite element methods (BP-FEMs), optimal convergence rates, superconvergence and optimal numerical stability can be achieved, respectively. Several interesting samples of 3D blending surfaces are provided, to display the remarkable advantages of the proposed approaches in this paper: unique solutions of blending surfaces, optimal blending surfaces in minimum energy, ease in handling the complicated boundary constraint conditions, and less CPU time and computer storage needed. This paper and Li (J. Comput. Math. 16 (1998) 457–480; J. Comput. Appl. Math.) provide a foundation of blending surfaces by PDE solutions, a new trend of computer geometric design.

Boundary penalty finite element methods for blending surfaces — II: biharmonic equations

Abstract In this paper the biharmonic equations are discussed, and the boundary penalty finite methods (BP-FEMs) using piecewise cubic Hermite elements are chosen to seek their approximate solutions, satisfying the normal derivative and periodical boundary conditions. Theoretical analysis is made to discover that when the penalty power σ=2,3 ( or 4) and 0

The collocation Trefftz method for biharmonic equations with crack singularities

Abstract The purpose of this paper is to extend the boundary approximation method proposed by Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], i.e. the collocation Trefftz method called in this paper, for biharmonic equations with singularities. First, this paper derives the Green formulas for biharmonic equations on bounded domains with a non-smooth boundary, and corner terms are developed. The Green formulas are important to provide all the exterior and interior boundary conditions which will be used in the collocation Trefftz method. Second, this paper proposes three crack models (called Models I, II and III), and the collocation Trefftz method provides their most accurate solutions. In fact, Models I and II resemble Motzs problem in Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], and Model III with all the clamped boundary conditions originated from Schiff et al. [The mathematics of finite elements and applications III, 1979]. Moreover, effects on d 1 of different boundary conditions are investigated, and a brief analysis of error bounds for the collocation Trefftz method is made. Since accuracy of the solutions obtained in this paper is very high, they can be used as the typical models in testing numerical methods. The computed results show that as the singularity models, Models I and II are superior to Model III, because more accurate solutions can be obtained by the collocation Trefftz method.

A radial basis collocation method for Hamilton-Jacobi-Bellman equations

In this paper we propose a semi-meshless discretization method for the approximation of viscosity solutions to a first order Hamilton-Jacobi-Bellman (HJB) equation governing a class of nonlinear optimal feedback control problems. In this method, the spatial discretization is based on a collocation scheme using the global radial basis functions (RBFs) and the time variable is discretized by a standard two-level time-stepping scheme with a splitting parameter @q. A stability analysis is performed, showing that even for the explicit scheme that @q=0, the method is stable in time. Since the time discretization is consistent, the method is also convergent in time. Numerical results, performed to verify the usefulness of the method, demonstrate that the method gives accurate approximations to both of the control and state variables.

Superconvergence of solution derivatives for the Shortley-Weller difference approximation of Poisson's equation. Part I: smoothness problems

The finite difference method (FDM) using the Shortley-Weller approximation can be viewed as a special kind of the finite element methods (FEMs) using the piecewise bilinear and linear functions, and involving some integration approximation. When u ∈ C3(S) (i.e., u ∈ C3,0(S)) and f ∈ C2(S), the superconvergence rate O(h2) of solution derivatives in discrete H1 norms by the FDM is derived for rectangular difference grids, where h is the maximal mesh length of difference grids used, and the difference grids are not confined to be quasiuniform. Comparisons are made on the analysis by the maximum principle and the FEM analysis, conversions between the FDM and the linear and bilinear FEMs are discussed, and numerical experiments are provided to support superconvergence analysis made.

Global superconvergence of simplified hybrid combinations for elliptic equations with singularities, I. Basic theorem

Abstract.We consider a non-overlapping domain decomposition method for diffusion-reaction problems which is known to converge strongly from previous work. We derive an a posteriori estimate which bounds the errors on the subdomains by the difference of traces of the subdomain solutions. If the domain decomposition method is discretized by finite elements we can adapt the techniques of the usual a posteriori error analysis for finite elements to get an a posteriori estimate for the discrete subdomain solutions.