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SYMPOSIUM MINISYMPOSIA

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FIRST SOUTHERN SYMPOSIUM ON COMPUTING

December 4-5, 1998
University of Southern Mississippi
Hattiesburg, Mississippi


ABSTRACT

Assessing Finite Element Solutions to the Stochastic Sturm-Liouville Equation

M. Eckhoff and Joseph Kolibal

Stochastic partial differential equations (SPDEs) have stimulated interest primarily in the theory of SPDE's, addressing statistical properties of the solution, however their application to the solution of problems remains substantially incomplete and numerical techniques for white-noise driven PDEs remain underdeveloped.

This study addresses numerical issues regarding the formulation and solution of the stochastic Sturm-Liouville equation

- Ñ·(p Ñu) + qu = x    in G        p(x) > 0,        q(x) ³ 0
(1)
in a finite element framework. Here, G is an open, bounded subset of Rd, u|G = 0, and the spatial white noise is given by:

x º d w
x1 ¼xd
 Î H-d(G)  .
where w Î C(G) is a d-parameter Wiener process on G. The regularity of the solution u Î H2-d(G) depends on d and the smoothness of G. It is known that, with G sufficiently smooth, u satisfies decreasing degrees of Hölder continuity for d £ 3. Several questions arise when formulating traditional numerical solutions for (1). We investigate the following methods for (deterministic) elliptic equations:

  1. Ritz: minimize the functional

    J[v] = 1
    2
    		 ó
    õ

    G 
    		 [ p Ñv ·Ñv + qv2 - 2vx]  ,        v Î H01(G)  .
    		(2)
  2. Bubnov-Galerkin: find u Î H1(G) such that

    ó
    õ

    G 
    		 [p Ñu ·Ñv + q u v] = ó
    õ

    G 
    		 vx ,               "v Î H01(G)  .
    		(3)
  3. Petrov-Galerkin: find u Î H0(G) such that

    ó
    õ

    G 
    		 u[-Ñ·(p Ñv) + qv] = ó
    õ

    G 
    		 vx ,            "v Î H02(G)  .
    		(4)

In each case, the approximate solution uh resides in a trial space Vh = span {f1, ¼, fn}. For the Galerkin methods, there is also a test space Th = span {y1, ¼, yn}. When x Î L2(G) is deterministic, (2) and (3) produce identical results for Th = Vh. The optimality of the numerical solutions is assessed (the Bubnov formulation gives optimal result when u is differentiable); however, the loss of differentiability can require shifting all derivatives. In addition, problems develop when white-noise forcing is applied in the Ritz method, where J[v] is a random number, requiring deeper interpretation of the variational approach.


Getting More Information

To obtain more information about the meeting send e-mail to: fscc98@pax.st.usm.edu.

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