Large-Scale Scientific Computing

This paper analyzes the discrete energy laws associated with first-order system least-squares (FOSLS) discretizations of timedependent partial differential equations. Using the heat equation and the time-dependent Stokes’ equation as examples, we discuss how accurately a FOSLS finite-element formulation adheres to the underlying energy law associated with the physical system. Using regularity arguments involving the initial condition of the system, we are able to give bounds on the convergence of the discrete energy law to its expected value (zero in the examples presented here). Numerical experiments are performed, showing that the discrete energy laws hold with order O h2p, where h is the mesh spacing and p is the order of the finite-element space. Thus, the energy law conformance is held with a higher order than the expected, O (h), convergence of the finite-element approximation. Finally, we introduce an abstract framework for analyzing the energy laws of general FOSLS discretizations.

[1]  Fred C. Schweppe,et al.  Uncertain dynamic systems , 1973 .

[2]  J. Ball.  OPTIMIZATION—THEORY AND APPLICATIONS Problems with Ordinary Differential Equations (Applications of Mathematics, 17) , 1984 .

[3]  I. P. Natanson,et al.  Theory of Functions of a Real Variable , 1955 .

[4]  Vladimir M. Veliov,et al.  High-Order Approximations to Nonholonomic Affine Control Systems , 2009, LSSC.

[5]  R. Vinter,et al.  When Is the Maximum Principle for State Constrained Problems Nondegenerate , 1994 .

[6]  Eric Walter,et al.  Ellipsoidal parameter or state estimation under model uncertainty , 2004, Autom..

[7]  A. A. Samarskii,et al.  Computational heat transfer , 1995 .

[8]  F. Lubyshev,et al.  Approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and states and with controls in the coefficients multiplying the highest derivatives , 2016 .

[9]  Fernando A. C. C. Fontes,et al.  Normality and Nondegeneracy for Optimal Control Problems with State Constraints , 2015, J. Optim. Theory Appl..

[10]  A. Manapova,et al.  On some optimal control problems and their finite difference approximations and regularization for quasilinear elliptic equations with controls in the coefficients , 2007 .

[11]  Vladimir M. Veliov,et al.  Set-Valued Analysis and Differential Inclusions , 1993 .

[12]  G. T. McAllister.  Difference methods for a nonlinear elliptic system of partial differential equations , 1966 .

[13]  A. Kurzhanski,et al.  On the Theory of Trajectory Tubes — A Mathematical Formalism for Uncertain Dynamics, Viability and Control , 1993 .

[14]  Pravin Varaiya,et al.  Dynamics and Control of Trajectory Tubes , 2014 .

[15]  Numerical solution of optimization problems for semi-linear elliptic equations with discontinuous coefficients and solutions , 2016 .

[16]  Internal ellipsoidal estimates for bilinear systems under uncertainty , 2016 .

[17]  M. L. Chambers.  The Mathematical Theory of Optimal Processes , 1965 .

[18]  Stanislav Mazurenko,et al.  A differential equation for the gauge function of the star-shaped attainability set of a differential inclusion , 2012 .

[19]  Approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions and with control in matching boundary conditions , 2014 .

[20]  F. Lubyshev.  Finite difference approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions , 2012 .

[21]  A. Kurzhanski,et al.  Ellipsoidal Calculus for Estimation and Control , 1996 .