High-order adaptive finite element-singly implicit Runge-Kutta methods for parabolic differential equations

We describe an adaptive mesh refinement finite element method-of-lines procedure for solving one-dimensional parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise hierarchical polynomial basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A modified SIRK formulation eliminates a linear systems solution that is required by the traditional SIRK formulation and leads to a new reduced-order interpolation formula. Stability and temporal error estimation techniques allow acceptance of approximate solutions at intermediate stages, yielding increased efficiency when solving partial differential equations. A priori energy estimates of the local discretization error are obtained for a nonlinear scalar problem. A posteriori estimates of local spatial discretization errors, obtained by order variation, are used with the a priori error estimates to control the adaptive mesh refinement strategy. Computational results suggest convergence of the a posteriori error estimate to the exact discretization error and verify the utility of the adaptive technique.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-90-0194; the U.S. Army Research Office under Contract Number DAAL 03-91-G-0215; by the National Science Foundation under Grant Number CDA-8805910; and by a grant from the Committee on Research, Tulane University.     

A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems

Summary Consider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewiseP ^th -degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degreep+1 that vanish at element ends. Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 90-0194; by the U.S. Army Research Office under Contract Number DAAL03-91-G-0215; and by the National Science Foundation under Institutional Infrastructure Grant Number CDA-8805910     

On conforming finite element methods for the inhomogeneous stationary Navier-Stokes equations

Summary We consider the stationary Navier-Stokes equations, written in terms of the primitive variables, in the case where both the partial differential equations and boundary conditions are inhomogeneous. Under certain conditions on the data, the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. A conforming finite element method is presented and optimal estimates for the error of the approximate solution are proved. In addition, the convergence properties of iterative methods for the solution of the discrete nonlinear algebraic systems resulting from the finite element algorithm are given. Numerical examples, using an efficient choice of finite element spaces, are also provided.Supported, in part, by the U.S. Air Force Office of Scientific Research under Grant No. AF-AFOSR-80-0083Supported, in part, by the same agency under Grant No. AF-AFOSR-80-0176-A. Both authors were also partially supported by NASA Contract No. NAS1-15810 while they were in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23665, USA     

Numerical methods for reaction-diffusion problems with non-differentiable kinetics

Summary We consider a class of steady-state semilinear reaction-diffusion problems with non-differentiable kinetics. The analytical properties of these problems have received considerable attention in the literature. We take a first step in analyzing their numerical approximation. We present a finite element method and establish error bounds which are optimal for some of the problems. In addition, we also discuss a finite difference approach. Numerical experiments for one- and two-dimensional problems are reported.Dedicated to Ivo Babuka on his sixtieth birthdayResearch partially supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under Grant Number AFOSR 85-0322     

On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver     

Locking effects in the finite element approximation of elasticity problems

Summary We consider the finite element approximation of the 2D elasticity problem when the Poisson ratiov is close to 0.5. It is well-known that the performance of certain commonly used finite elements deteriorates asv0, a phenomenon calledlocking. We analyze this phenomenon and characterize the strength of the locking androbustness of varioush-version schemes using triangular and rectangular elements. We prove that thep-andh-p versions are free of locking with respect to the error in the energy norm. A generalization of our theory to the 3D problem is also discussed.The work of this author was supported in part by the Office of Naval Research under Naval Research Grant N00014-90-J-1030The work of this author was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, U.S. Air Force, under grant AFOSR 89-0252     

The treatment of nonhomogeneous Dirichlet boundary conditions by thep-version of the finite element method

Summary The paper addresses the problem of the implementation of nonhomogeneous essential Dirichlet type boundary conditions in thep-version of the finite element method.Partially supported by the Office of Naval Research under Grant N-00014-85-K-0169Research partially supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0322     

Solution manifolds and submanifolds of parametrized equations and their discretization errors

Summary The paper concerns solution manifolds of nonlinear parameterdependent equations (1)F(u, )=y₀ involving a Fredholm operatorF between (infinite-dimensional) Banach spacesX=Z× andY, and a finitedimensional parameter space . Differntial-geometric ideas are used to discuss the connection between augmented equations and certain onedimensional submanifolds produced by numerical path-tracing procedures. Then, for arbitrary (finite) dimension of , estimates of the error between the solution manifold of (1) and its discretizations are developed. These estimates are shown to be applicable to rather general nonlinear boundaryvalue problems for partial differential equations.This work was in part supported by the U.S. Air Force Office of Scientific Research under Grant 80-0176, the National Science Foundation under Grant MCS-78-05299, and the Office of Naval Research under Contract N-00014-80-C-0455     

Finite element methods for semilinear elliptic stochastic partial differential equations

We study finite element methods for semilinear stochastic partial differential equations. Error estimates are established. Numerical examples are also presented to examine our theoretical results. This research is supported by Air Force Office of Scientific Research under the grant number FA9550-05-1-0133 and 985 Project of Jilin University.     

On the stability of bilinear-constant velocity-pressure finite elements

Summary An analysis of the Babuka stability of bilinear/constant finite element pairs for viscous flow calculations is given. An unstable mode not of the checkerboard type is given for which the stability constant turns out to beO(h). Thus, the indicated spaces are not stable in general for numerical calculation.Work supported by U.S. Air Force Office of Scientific Research under grant AF-AFOSR-82-0213     

Problems and methods in partial differential equations

Summary The method of singularities is used to solve theCauchy problem for simple hyperbolic partial differential equations, namely, the wave equation and the damped wave equation. The representation formula for the solution of theCauchy problem is written in terms of finite parts and logarithmic parts of certain divergent integrals. A process of analytic continuation is also used to solve theCauchy problems under consideration. However, to obtain explicitly the representation formulas for the solutions, one must actually perform the analytic continuation. It is shown that this is best achieved by making use of finite and logarithmic parts. Simple examples were purposely chosen so as to show that consideration of finite and logarithmic parts is naturally unavoidable and ? in the very nature of things ?. To Enrico Bompiani on his scientific Jubilee. This work was sponsored in part by the Air Force Office of Scientific Research of the Air Research and Development Command, United States Air Force, through its European Office.     

On a singular differential operator

Summary Study of the properties of a singular differential operator in one independent variable. Applications to ordinary and partial differential equations. To Giovanni Sansone on his 70^th birth day. This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under Contract No. 49 (638)-228.     

A partially observed control problem for Markov chains

A finite state, continuous time Markov chain is considered and the solution to the filtering problem given when the observation process counts the total number of jumps. The Zakai equation for the unnormalized conditional distribution is obtained and the control problem discussed in separated form with this as the state. A new feature is that, because of the correlation between the state and observation process, the control parameter appears in the diffusion coefficient which multiplies the Poisson noise in the Zakai equation. By introducing a Gâteaux derivative the minimum principle, satisfied by an optimal control, is derived. If the optimal control is Markov, a stochastic integrand can be obtained more explicitly and new forward and backward equations satisfied by the adjoint process are obtained.This research was partially supported by NSERC Grant A7964, the Air Force Office of Scientific Research, United States Air Force, under Contract AFOSR-86-0332, and the U.S. Army Research Office under Contract DAAL03-87-0102.     

Linear complexity algorithms for semiseparable matrices

Linear complexity algorithms are derived for the solution of a linear system of equations with the coefficient matrix represented as a sum of diagonal and semiseparable matrices. LDU-factorization algorithms for such matrices and their inverses are also given. The case in which the solution can be efficiently update is treated separately.This work was supported in part by the U.S. Army Research Office, under Contract DAAG29-83-K-0028, and the Air Force Office of Scientific Research, Air Force Systems Command under Contract AF83-0228.     

Spherical means in spaces of constant curvature

Summary Extension of theorems ofF. Riesz on subharmonic functions to spaces of constant curvature by the use of hyperbolic partial differential equations. To Enrico Bompiani on his scientific Jubiles This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under ontract No. AF 49 (638)–228.     

Some mathematical problems in computer vision

Several interesting mathematical problems arising in computer vision are discussed. Computer vision deals with image understanding at various levels. At the low level, it addresses issues like segmentation, edge detection, planar shape recognition and analysis. Classical results on differential invariants associated to planar curves are relevant to planar object recognition under partial occlusion, and recent results concerning the evolution of closed planar shapes under curvature controlled diffusion have found applications in shape decomposition and analysis. At higher levels, computer vision problems deal with attempts to invert imaging projections and shading processes toward depth recovery, spatial shape recognition and motion analysis. In this context, the recovery of depth from shaded images of objects with smooth, diffuse surfaces require the solution of nonlinear partial differential equations. Here results on differential equations, as well as interesting results from low-dimensional topology and differential geometry are the necessary tools of the trade. We are still far from being able to equip our computers with brains capable to analyze and understand the images that can easily be acquired with camera-eyes; however the research effort in this area often calls for both classical and recent mathematical results.This work was supported in part by NSF grant DMS-8811084, Air Force Office of Scientific Research Grant AFOSR-90-0024, and the Army Research Office DAAL03-91-G-0019, and by the Technion Fund for Promotion of Research.     

Fast algorithms for the integral equations of the inverse scattering problem

The Gelfand-Levitan and Marchenko equations of inverse scattering theory are integral equations with Toeplitz and Hankel kernels respectively. It is shown that these facts can be used to reduce the integral equations to differential equations which can be solved with an order of magnitude less computation than generally envisaged.This work was supported by the Army Research Office under Contract DAAG29-77-C-0042, by the Air Force Office of Scientific Research, Air Force Systems Command, under Contract AF44-620-74-C-0068 and the Australian Research Grants Committee.     

On the computation of the optimal cost function for discrete time Markov models with partial observations

We consider several applications of two state, finite action, infinite horizon, discrete-time Markov decision processes with partial observations, for two special cases of observation quality, and show that in each of these cases the optimal cost function is piecewise linear. This in turn allows us to obtain either explicit formulas or simplified algorithms to compute the optimal cost function and the associated optimal control policy. Several examples are presented.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-86-0029, in part by the National Science Foundation under Grant ECS-8617860, in part by the Advanced Technology Program of the State of Texas, and in part by the DoD Joint Services Electronics Program through the Air Force Office of Scientific Research (AFSC) Contract F49620-86-C-0045.     

Reflection principles for linear elliptic second order partial differential equations with constant coefflcients

Summary Reflection principles, analogous to the classicalSchwarz reflection principle for harmonic functions, are obtained for solutions of linear elliptic second order partial differential equations with constant coefficients. The boundary conditions employed are supposed to be satisfied in a limiting sense only, and do not require (a priori) the existence of derivatives on the boundary. To Mauro Picone on his 70^th birth day. This research was supported in part by the United States Air Force under Contract N^o. AF(600)-573 — monitored by the Office of Scientific Research, Air Research and Development Command. The work of this author was sponsored by the Office of Ordnance Research, U.S. Army, under the Contract DA-36-034-ORD-1486.     

On the stationary semiconductor equations arising in modeling an LBIC technique

In this paper by using anL estimate for elliptic equations, we study the well-posedness of the stationary semiconductor equations arising from modeling a nondestructive testing technique LBIC. It is shown that when the extra source term is small, the system has a unique weak solution, and the solution is continuously dependent on this source term. The validity of an approximate model derived for the study of the inverse problem is established. The existence result is then extended to the case of constant mobilities without the assumption on the size of the source term.Parts of this work were completed while the authors were members of the Center for Applied Mathematical Sciences at the University of Southern California, and was supported by Air Force Office of Scientific Research Grant AFOSR-90-0091.