Elliptic Equations of Higher Stochastic Order

Abstract

This paper discusses analytical and numerical issues related to elliptic equations with random coefficients which are generally nonlinear functions of white noise. Singularity issues are avoided by using the Ito-Skorohod calculus to interpret the interactions between the coefficients and the solution. The solution is constructed by means of the Wiener Chaos (Cameron-Martin) expansions. The existence and uniqueness of the solutions are established under rather weak assumptions, the main of which requires only that the expectation of the highest order (differential) operator is a non-degenerate elliptic operator. The deterministic coefficients of the Wiener Chaos expansion of the solution solve a lower-triangular system of linear elliptic equations (the propagator). This structure of the propagator insures linear complexity of the related numerical algorithms. Using the lower triangular structure and linearity of the propagator, the rate of convergence is derived for a spectral/hp finite element approximation. The results of related numerical experiments are presented.

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Document Details

Document Type
Technical Report
Publication Date
Jan 01, 2009
Accession Number
ADA597555

Entities

People

  • B. L. Rozovskii
  • S. V. Lototsky
  • X. Wan

Organizations

  • Brown University

Tags

Communities of Interest

  • C4I

DTIC Thesaurus Topics

  • Algorithms
  • Applied Mathematics
  • Computational Complexity
  • Computations
  • Convergence
  • Equations
  • Finite Element Analysis
  • Inequalities
  • Mathematics
  • Noise
  • Numbers
  • Numerical Analysis
  • Random Variables
  • Real Numbers
  • Theorems
  • Two Dimensional
  • White Noise

Fields of Study

  • Mathematics

Readers

  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Linear Algebra
  • Mathematical Modeling and Probability Theory.