Subordinated Gaussian Random Fields in Elliptic Partial Differential Equations

TL;DR

This paper introduces a novel subordination method to generate discontinuous Le9vy-type random fields for elliptic PDEs in two-dimensional heterogeneous media, proving existence, uniqueness, and providing numerical validation.

Contribution

It develops a new subordination approach for two-dimensional Le9vy-type fields, extending modeling capabilities for complex subsurface flows with discontinuities.

Findings

Existence and uniqueness of solutions are established.

A Monte Carlo finite element method validates the theoretical results.

The approach effectively models spatial discontinuities in heterogeneous media.

Abstract

To model subsurface flow in uncertain heterogeneous\ fractured media an elliptic equation with a discontinuous stochastic diffusion coefficient - also called random field - may be used. In case of a one-dimensional parameter space, L\'evy processes allow for jumps and display great flexibility in the distributions used. However, in various situations (e.g. microstructure modeling), a one-dimensional parameter space is not sufficient. Classical extensions of L\'evy processes on two parameter dimensions suffer from the fact that they do not allow for spatial discontinuities. In this paper a new subordination approach is employed to generate L\'evy-type discontinuous random fields on a two-dimensional spatial parameter domain. Existence and uniqueness of a (pathwise) solution to a general elliptic partial differential equation is proved and an approximation theory for the diffusion coefficient and the corresponding solution provided. Further, numerical examples using a Monte Carlo approach on a Finite Element discretization validate our theoretical results.