Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients

Abstract

This paper proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients, under the finite noise assumption. First, the stochastic discontinuous Galerkin method represents the stochastic solution in a Galerkin framework. Second, the Monte Carlo discontinuous Galerkin method samples the coefficients by a Monte Carlo approach. Both methods discretize the differential operators by the class of interior penalty discontinuous Galerkin methods. Error analysis is obtained. Numerical results show the sensitivity of the expected value and variance with respect to the penalty parameter of the spatial discretization.

Keywords:

• stochastic discontinuous Galerkin
• finite noise
• stochastic elliptic equation
• Monte Carlo simulation
• a priori error estimate

2010 AMS Subject Classifications:

• 65N30
• 65N15
• 65C05
• 65C20
• 65N12