Daoqi Yang

Department of Mathematics
Wayne State University
Detroit, Michigan 48202
yang@math.wayne.edu
Phone: (313)577-2491
Fax: (313)577-7596

Abstracts of Daoqi Yang's Publications

The mixed finite element methods with moving grids for parabolic problems, Mathematica Numerica Sinica, 10(1988), pp. 266-271. (Mathematical Reviews 90g: 65153)

Four fully discrete mixed finite element methods with moving grids are presented for approximating the solution to second order nonlinear parabolic problems. It is shown that these methods have the advantages of the mixed finite elements and moving finite elements. Error estimates are established. These methods are proved to have the optimal convergent orders in some sense. ---- from Mathematical Reviews.

Multistep schemes for mixed finite element methods for second order parabolic problems, Journal of Qingdao University, 2 (1989), pp. 56-62. (Zbl. Math. 788: 65099)

Several linear multistep schemes for mixed finite element methods are discussed for approximating the solution and its gradient for second order parabolic partial differential equations. These fully discrete schemes, which have orders in time O (k^m), m = 1, 2, 3, 4, where k is the time step, enable one to apply large time steps in order to reduce computational work, while maintaining the same rates of convergence. Error estimates show that these schemes have optimal rates of convergence in the L² norm sense. ---- from Zbl. Math.

Mixed methods with dynamic finite-element spaces for miscible displacement in porous media, Journal of Computational and Applied Mathematics, 30 (1990), pp. 313-328. (Mathematical Reviews 91k: 76114)

The author considers the approximation of the miscible displacement equations, a system consisting of an elliptic and a parabolic equation coupled in flux terms, by mixed finite elements in space and finite differences in time. The nonlinearity is handled either by explicit linearization and the backward Euler method, or by extrapolation from two old time levels and the Crank-Nicolson method. The author allows for a change of the approximation spaces from time level to time level (e.g. caused by regridding). He shows that the optimal L²-error estimates still holds true if the number of changes of approximation is bounded. ---- from Mathematical Reviews.

A characteristic mixed method with dynamic finite element space for convection-dominated diffusion problems, Journal of Computational and Applied Mathematics, 43 (1992), pp. 343-353. (Mathematical Reviews 93k: 65079)

This very attractive paper gives convergence results and a priori estimates for a first-order-in-time method of solution for a general parabolic equation with dominant convection term.
The method starts from an operator splitting where the convection term is isolated in order to apply a linear characteristic method; thus the diffusion step is formulated using the function to be determined and its gradient as unknowns. The variational formulation is given in the proper Sobolev spaces and a dynamic (time-dependent) finite element scheme is applied. The advantages of obtaining the function and its gradient simultaneously is emphasized, which allows one to use large time steps, grid refinements and/or basis function adjustments.
Various references are given in which numerical applications prove the efficiency of the method in cases where localized phenomena such as fronts, shocks and boundary layers appear in the solution.
Extensions to cases where the diffusion coefficient and the right-hand side in the equation are solution-dependent as well as extensions to the three-dimensional case are still under development. ---- from Mathematical Reviews.

Grid modification for the wave equation with attenuation, Numerische Mathematik 67 (1994), pp. 391-401. (Mathematical Reviews 95b: 65113)

The second-order hyperbolic partial differential wave equation with attenuation due to linear friction is approximated by a new mixed finite element method with a dynamic function space, which allows the use of different grids and different basis functions at different times when necessary. This method enables one to track sharp moving wave fronts more efficiently and accurately and provides the ability to approximate the displacement and stresses simultaneously and obtain high-order approximations of the stresses. A priori error estimates with optimal convergent rates are established and can provide a tool to locally refine and coarsen grids and/or adjust basis functions at different times. Unconditional stability is also proved for this method. ---- from Mathematical Reviews.
postscript file

Different domain decompositions at different times for capturing moving local phenomena, Journal of Computational and Applied Mathematics, 59 (1995), pp. 39-48. (Mathematical Reviews 96f:65130)

The objective of this paper is to propose a numerical method for parabolic problems which allows one to use different domain decompositions at different time levels when necessary. It uses an implicit finite element method. This method is illustrated by numerical experiments and will be very useful for researchers working in this area. ---- from Mathematical Reviews.
postscript file

Grid modification for second order hyperbolic problems, Mathematics of Computation, 64 (1995), pp. 1495-1509. (Mathematical Reviews 95m:65173)

A family of Galerkin finite element methods is presented to accurately and efficiently solve the wave equation that includes sharp propagating wave fronts. The new methodology involves different finite element discretizations at different time levels; thus, at any time level, relatively coarse grids can be applied in regions where the solution changes smoothly while finer grids can be employed near wave fronts. The change of grid from time step to time step need not be continuous, and the number of grid points at different time levels can be arbitrarily different. The formulation is applicable to general second order hyperbolic equations. Stability results are proved and a-priori error estimates are established for several boundary conditions. Our error estimates consist of three parts: the time finite difference discretization error, the spatial finite element discretization error, and the error due to the projections of the approximated solution from old grids onto new grids.
postscript file

A parallel iterative nonoverlapping domain decomposition procedure for elliptic problems, IMA Journal on Numerical Analysis, 16(1996) 75-91.

A parallel iterative nonoverlapping domain decomposition method is proposed and analyzed for elliptic problems. Each iteration in this method contains two steps. In the first step, at the interface of two subdomains, one subdomain problem requires that Dirichlet data be passed to it from the previous iteration level, while the other subdomain problem requires Neumann data be passed to it. In the second step, we interchange the types of data passing at the interface of the two subdomains. This domain decomposition method is suitable for parallel processing with coarse granularity. Convergence analysis is demonstrated at the differential level by Hilbert space techniques. Numerical results are provided to confirm the convergence theory.
postscript file

Jim Douglas, Jr. and Daoqi Yang, Numerical experiments of a nonoverlapping domain decomposition method for partial differential equations, in: D. Griffiths and G. A. Watson, Eds., Numerical Analysis: A. R. Mitchell 75th Birthday Volume, World Scientific Publishing Co., Singpore, 1996, pp. 85-97.

We present numerical experiments for a nonoverlapping domain decomposition method with interface relaxation for general selfadjoint and non-selfadjoint elliptic problems in two-dimensions. The procedure contains two steps in each full iteration. The transmission condition on the interface is taken to be Dirichlet in the first step and Neumann in the second. However, in the presence of interior subdomains, an average mechanism is introduced at each cross point to update the value at these points right after the Neumann sweep. Numerical examples show the fast convergence and accuracy of the method.
postscript file

A Parallel Nonoverlapping Schwarz Domain Decomposition Algorithm for Elliptic Partial Differential Equations, Proceedings of the Eighth SIAM Conference on Parallel Processing for Scientific Computing (Minneapolis, March 14-17, 1997), Proceedings in Applied Mathematics 94, Eds., M. Heath, V. Torczon, G. Astfalk, P. Bjorstad, A. H. Karp, C. H. Koebel, V. Kumar, R. E. Lucas, L. T. Watson, and D. E. Womble, SIAM, Philadelphia, PA, 1997.

A parallel nonoverlapping Schwarz algorithm based on a finite element discretization is applied to convection diffusion problems with fronts and layers. At inter-domain boundaries, a Robin type boundary condition is weakly imposed into the finite element variational form. Grid refinement is made on subdomains that contain fronts and layers and coarse grids are applied on subdomains in which the solution changes slowly and smoothly. In order to balance loads among different processors, the sizes of subdomains are chosen such that each subdomain problem has approximately the same number of degrees of freedom. Message passing parallel implementations on an nCUBE2 machine are conducted to show the efficiency and accuracy of the method.
postscript file

An Iterative Perturbation Method for the Pressure Equation in the Simulation of Miscible Displacement in Porous Media, SIAM Journal on Scientific Computing, 19(1998) 893 - 911. (jointly with Ping Lin)

The miscible displacement problem in porous media is modeled by a nonlinear coupled system of two partial differential equations: the pressure-velocity equation and the concentration equation. An iterative perturbation procedure is proposed and analyzed for the pressure-velocity equation, which is capable of producing as accurate velocity approximation as the mixed finite element method, and requires the solution of symmetric positive definite linear systems. Only the velocity variable is involved in the linear systems and the pressure variable is obtained by substitution. Babuska-Brezzi Trivially applying perturbation methods can only give an error $O(\epsilon)$, while our iterative scheme can improve the error to $O(\epsilon^m)$ at the $m$-th iteration level, where $\epsilon$ is a small positive number. Thus the convergence rate of our iterative procedure is $O(\epsilon)$, and consequently a small number of iterations are required. Theoretical convergence analysis and numerical experiments are presented to show the efficiency and accuracy of our method.
postscript file

Dynamic Domain Decomposition and Grid Modification for Parabolic Problems, Computers and Mathematics with Applications, 33(1997), 89-103.

A dynamic grid modification and domain decomposition method is given and analyzed for parabolic problems. This method allows one to apply different domain decompositions, and different grids and interpolation polynomials on the subdomains at different time levels when necessary. The procedure relies on an implicit Galerkin method in the subdomains and explicit flux calculation on the inter-domain boundaries. In addition, a dynamic finite element scheme is proposed and analyzed, which is applicable to general parabolic problems. These methods are well suited to large-scale time-dependent problems involving localized phenomena, such as sharp fronts or layers, which also change with time. Convergence and stability analyses in the L² norm are given. Numerical experiments are provided to check the performance of the methods and make comparison with other methods.
postscript file

A Parallel Iterative Domain Decomposition Algorithm for Elliptic Problems, Journal of Computational Mathematics, 16(1998) 141-151.

An iterative nonoverlapping domain decomposition procedure is proposed and analyzed for linear elliptic problems. At the interface of two subdomains, one subdomain problem requires that Dirichlet data be passed to it from the previous iteration level, while the other subdomain problem requires that Neumann data be passed to it. This procedure is suitable for parallel processing. A convergence analysis is established. Standard and mixed finite element methods are employed to give discrete versions of this domain decomposition algorithm. Numerical experiments are conducted to show the effectiveness of the method.
postscript file

John R. Rice, E. A. Vavalis and Daoqi Yang, Convergence analysis of a nonoverlapping domain decomposition method for elliptic PDEs, J. Comput. Appl. Math., 87(1997), 11-19.

In this study we analyze a non-overlapping domain decomposition method for the solution of elliptic Partial Differential Equation (PDE) problems. This domain decomposition method involves the solution of Dirichlet and Neumann PDE problems on each subdomain, coupled with smoothing operations on the interfaces of the subdomains. The convergence analysis of the method at the differential equation level is presented. The numerical results confirm the theoretical ones and exhibit computational efficiency of the method.
postscript file

A parallel grid modification and domain decomposition algorithm for local phenomena capturing and load balancing, Journal of Scienttific Computing, 12(1997) 99-117.

Lions' non-overlapping Schwarz domain decomposition method based on a finite difference discretization is applied to problems with fronts or layers. For the purpose of getting accurate approximation of the solution by solving small linear systems, grid refinement is made on subdomains that contain fronts and layers and uniform coarse grids are applied on subdomains in which the solution changes slowly and smoothly. In order to balance loads among different processors, we employ small subdomains with fine grids for rapidly-changing-solution areas, and big subdomains with coarse grids for slowly-changing-solution areas. Numerical implementations in the SPMD mode on an nCUBE2 machine are conducted to show the efficiency and accuracy of the method.

Simulation of Miscible Displacement in Porous Media by a modified Uzawa's algorithm Combined with a Characteristic Method, Computer Methods in Applied Mechanics and Engineering, 162(1998) 359-368.

Miscible displacement in porous media is modeled by a nonlinear coupled system of two partial differential equations: the pressure equation, which is elliptic, and the concentration equation, which is parabolic but usually convection-dominated. In this paper, we focus on some computational difficulties in the simulation of this problem. In order to obtain accurate approximation for velocity, we implement an iterative perturbation algorithm for the pressure equation with a full permeability tensor including strong heterogeneity and anisotropy. For handling the convection dominance over diffusion, we employ a modified method of characteristics for the concentration equation. Numerical simulation results show that the combination of the modified method of characteristics and the iterative perturbation algorithm is a good choice for the problem. In particular, the modified method of characteristics dramatically reduces numerical diffusion and non-physical oscillation in the approximate concentration, and the iterative perturbation algorithm leads to positive definite linear systems with velocity unknowns only and exhibits the accuracy of mixed finite element methods.
postscript file

An augmented Lagrangian mixed finite element scheme for saddle point problems, in: J. Wang, M. B. Allen III, B. M. Chen and T. Mathew, Editors, Iterative Methods in Scientific Computation, IMACS Series in Computational and Applied Mathematics, Vol 4, IMACS, New Brunswick, NJ, 1998, pp. 325-330.

An augmented Lagrangian iterative scheme for mixed finite element methods is proposed and analyzed in an abstract framework. This framework is based on the iterative schemes in the book: The Mathematical Theory of Finite Element Methods, by Brenner and Scott (Springer, 1994), but is more general and computationally efficient. The linear systems of algebraic equations obtained from this scheme are positive definite and have a smaller number of degrees of freedom than standard mixed finite element methods, and may have smaller condition numbers than penalty methods. Theoretical analysis and computational experiments both show that the scheme has very fast convergence; a few iterations are enough to reduce the iterative error to a prescribed precision.
postscript file

An Iterative Perturbation Method for Saddle Point Problems, IMA Journal of Numerical Analysis, 19(1999) 215-231.

An iterative perturbation method is presented for saddle point problems as an iterative scheme for mixed finite element methods. It can be viewed as an extension of augmented Lagrangian methods to second order elliptic equations with a reaction term. This iterative method can lead to positive definite systems and decouple the flux variable from the pressure variable. Convergence analyses are established at the differential and finite-dimensional levels. Various computational experiments for problems with continuous and discontinuous coefficients and full diffusion tensors are conducted to confirm the theoretical claims. These experiments are implemented in the object oriented programming paradigm using C++ to take the advantage of object oriented software engineering techniques such as encapsulation, inheritance, and polymorphism.
postscript file

A Parallel Nonoverlapping Schwarz Domain Decomposition Method for Elliptic Interface Problems, to appear.

We present a parallel nonoverlapping Schwarz domain decomposition method with interface relaxation for linear elliptic problems, possibly with discontinuities in the solution and its derivatives. This domain decomposition method can be characterized as: the transmission conditions at the interfaces of subdomains are taken to be Dirichlet at odd iterations and Neumann at even iterations. The convergence analysis of the iterative sequence of subdomain solutions is first established in the energy norm sense at the continuous level. Finite-dimensional discretization schemes, such as finite element approximation, finite element approximation with Lagrange multipliers and hybrid mixed finite element approximation, are then considered and analyzed. Numerical examples are provided to check the performance of this iterative procedure.

Dynamic Finite Element Methods for Second Order Parabolic Equations, to appear.

Dynamic finite element schemes are analyzed for second order parabolic problems. These schemes can employ different finite element spaces at different time levels in order to capture time-changing localized phenomena, such as moving sharp fronts or layers. The dynamically changing grids and interpolation polynomials are necessary and essential to many large-scale transient problems. Standard, characteristic, and mixed finite element methods with dynamic function spaces are considered for linear and nonlinear problems. The convergence results obtained in this paper are optimal and better than those published previously.

Stabilized Schemes for Mixed Finite Element Methods with Applications to Elasticity and Compressible Flow Problems, to appear.

Stabilized iterative schemes for mixed finite element methods are proposed and analyzed in two abstract formulations. The first one has applications to elliptic equations and incompressible fluid flow problems, while the second has applications to linear elasticity and compressible Stokes problems. Convergence theorems are demonstrated in abstract formulations; applications to individual physical problems are included. In contrast to standard mixed finite element methods, these stabilized schemes lead to positive definite linear systems of algebraic equations that have smaller condition numbers than penalty methods. Theoretical analysis and computational experiments both show that the stabilized schemes have very fast convergence; a few iterations are usually enough to reduce the iterative error to a prescribed precision. Numerical examples with continuous and discontinuous coefficients are presented.

Hilbert spaces and quatum mechanics, Wiley Encyclopedia of Electrical and Electronics Engineering, Vol 9, pp. 73-83. Editor, J. Webster, John Wiley & Sons, New York.

This article talks about the theory and methods of Hilbert spaces with applications to electrical and electronics engineering, especially to quatum mechanics.
Keywords: Classical integrable systems, integrable calssical field theories, quantization, algebraic structures, analytical structrues, quantum integrable systems.

A Nonoverlapping Subdomain Algorithm with Lagrange Multipliers and its Object Oriented Implementation for Interface Problems, in: Domain Decomposition Methods 10, Contemporary Mathematics, Vol 218, J. Mandel, C. Farhat, and X. Cai, Editors. American Mathematical Society, Providence, RI, 1998, pp 365-373.

A parallel nonoverlapping subdomain Schwarz alternating algorithm with Lagrange multipliers and interface relaxation is proposed for linear elliptic interface problems with discontinuities in the solution, its derivatives, and the coefficients. New features of the algorithm include that Lagrange multipliers are introduced on the interface and that it is used to solve equations with discontinuous solution. These equations have important applications to alloy solidification problems \cite{JCrank1984a} and immiscible flow of fluids with different densities and viscosities and surface tension. They do not fit into the Schwarz preconditioning and Schur complement frameworks since the solution is not in $H^1(\Omega),$ but is piecewise in $H^1(\Omega),$ where $\Omega$ is the physical domain on which the differential equations are defined. An expression for the optimal interface relaxation parameters is also given. Numerical experiments are conducted for a piecewise linear triangular finite element discretization in object oriented paradigm using C++. In this implementation, the class for subdomain solvers inherits from the class for grid triangulation and contains type bound procedures for forming stiffness matrices and solving linear systems. From software engineering point of view, features like encapsulation, inheritance, polymorphism and dynamic binding, make such domain decomposition algorithms an ideal application area of object oriented programming.
postscript file

Finite Elements for Elliptic Interface Problems with Strongly Discontinuous Coefficients and Solutions

An iterative finite element algorithm is proposed for numerically solving two-phase steady-state generalized Stefan interface problems with discontinuous solutions, conormal derivatives, and coefficients. This algorithm employs finite element methods and iteratively solves smaller subregion problems for each phase with good accuracy, and exchanges information at the interface to advance the iteration until convergence, following the idea of the Schwarz Alternating Method. The finite element grids in different phases do not have to match each other at the interface. Numerical experiments are performed to show the accuracy and efficiency of the algorithm for capturing discontinuities in the solutions and coefficients. One surprising property of the algorithm is that its accuracy does not deteriorate as the discontinuity in the coefficients becomes worse. That is, the accuracy remains the same for continuous problems and strongly discontinuous problems. Another surprising property is that its conditioning becomes better as the discontinuity gets worse. In other words, the stronger the discontinuity, the faster convergence. Numerical examples on matching and non-matching grids are given with coefficient discontinuity jumps in the order of 10³, 10⁵, 10¹⁰, 10⁵⁰, and 10¹⁰⁰.

An iterative hybridized mixed finite element method for elliptic interface problems with strongly discoefficients.

An iterative algorithm is proposed and analyzed based on a hybridized mixed finite element method for numerically solving two-phase generalized Stefan interface problems with strongly discontinuous coefficients. This algorithm iteratively solves small problems for each single phase with good accuracy and exchange information at the interface to advance the iteration until convergence, following the idea of Schwarz Alternating Methods. Error estimates are derived to show that this algorithm always converges provided that relaxation parameters are suitably chosen. Numerical experiments with matching and non-matching grids at the interface from different phases are performed to show the accuracy of the method for capturing strong discontinuities in the coefficients.

Click here to get to the begining of my home page.
Comments: yang@math.wayne.edu