Reduced Basis Methods for Partial Differential Equations : Evaluation of multiple non-compliant flux-type output functionals for a non-affine electrostatics problem

A method for rapid evaluation of flux-type outputs of interest from solutions to partial differential equations (PDEs) is presented within the reduced basis framework for linear, elliptic PDEs. The central point is a Neumann-Dirichlet equivalence that allows for evaluation of the output through the bilinear form of the weak formulation of the PDE. Through a comprehensive example related to electrostatics, we consider multiple outputs, a posteriori error estimators and empirical interpolation treatment of the non-affine terms in the bilinear form. Together with the considered Neumann-Dirichlet equivalence, these methods allow for efficient and accurate numerical evaluation of a relationship mu->s(mu), where mu is a parameter vector that determines the geometry of the physical domain and s(mu) is the corresponding flux-type output matrix of interest. As a practical application, we lastly employ the rapid evaluation of s-> s(mu) in solving an inverse (parameter-estimation) problem.

[1]  W. J. Gordon,et al.  Construction of curvilinear co-ordinate systems and applications to mesh generation , 1973 .

[2]  Sigal Gottlieb,et al.  Spectral Methods , 2019, Numerical Methods for Diffusion Phenomena in Building Physics.

[3]  John M. Lee.  Introduction to Smooth Manifolds , 2002 .

[4]  Robert McOwen,et al.  Partial differential equations : methods and applications , 1996 .

[5]  Yvon Maday,et al.  A reduced basis element method for the steady stokes problem , 2006 .

[6]  J. Hesthaven,et al.  Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations , 2007 .

[7]  Anthony T. Patera,et al.  A GENERAL OUTPUT BOUND RESULT: APPLICATION TO DISCRETIZATION AND ITERATION ERROR ESTIMATION AND CONTROL , 2001 .

[8]  Ivo Babuška,et al.  The p -version of the finite element method for the elliptic boundary value problems with interfaces , 1992 .

[9]  D. Greenfield,et al.  Three-dimensional electrostatic field calculation with effective algorithm of surface charge singularities treatment based on the Fichera's theorem , 2004 .

[10]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

[11]  D. Rovas,et al.  Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods , 2002 .

[12]  M. Grepl.  Reduced-basis approximation a posteriori error estimation for parabolic partial differential equations , 2005 .

[13]  E. Kreyszig.  Introductory Functional Analysis With Applications , 1978 .

[14]  A. Quarteroni,et al.  Numerical solution of parametrized Navier–Stokes equations by reduced basis methods , 2007 .

[15]  A. Quarteroni,et al.  Approximation results for orthogonal polynomials in Sobolev spaces , 1982 .

[16]  N. Nguyen,et al.  EFFICIENT REDUCED-BASIS TREATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 2007 .

[17]  A. Patera.  A spectral element method for fluid dynamics: Laminar flow in a channel expansion , 1984 .

[18]  Anthony T. Patera,et al.  Reduced basis approximation and a posteriori error estimation for a Boltzmann model , 2007 .

[19]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[20]  Yvon Maday,et al.  Polynomial approximation of some singular functions , 1991 .

[21]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .