Cody Lorton

Xiaobing Feng, Cody Lorton

This paper develops and analyzes an efficient Monte Carlo interior penalty discontinuous Galerkin (MCIP-DG) method for elastic wave scattering in random media. The method is constructed based on a multi-modes expansion of the solution of the governing random partial differential equations. It is proved that the mode functions satisfy a three-term recurrence system of partial differential equations (PDEs) which are nearly deterministic in the sense that the randomness only appears in the right-hand side source terms, not in the coefficients of the PDEs. Moreover, the same differential operator applies to all mode functions. A proven unconditionally stable and optimally convergent IP-DG method is used to discretize the deterministic PDE operator, an efficient numerical algorithm is proposed based on combining the Monte Carlo method and the IP-DG method with the $LU$ direct linear solver. It is shown that the algorithm converges optimally with respect to both the mesh size $h$ and the sampling number $M$, and practically its total computational complexity is only amount to solving very few deterministic elastic Helmholtz equations using the $LU$ direct linear solver. Numerically experiments are also presented to demonstrate the performance and key features of the proposed MCIP-DG method.

Xiaobing Feng, Junshan Lin, Cody Lorton

This paper develops an efficient Monte Carlo interior penalty discontinuous Galerkin method for electromagnetic wave propagation in random media. This method is based on a multi-modes expansion of the solution to the time-harmonic random Maxwell equations. It is shown that each mode function satisfies a Maxwell system with random sources defined recursively. An unconditionally stable IP-DG method is employed to discretize the nearly deterministic Maxwell system and the Monte Carlo method combined with an efficient acceleration strategy is proposed for computing the mode functions and the statistics of the electromagnetic wave. A complete error analysis is established for the proposed multi-modes Monte Carlo IP-DG method. It is proved that the proposed method converges with an optimal order for each of three levels of approximations. Numerical experiments are provided to validate the theoretical results and to gauge the performance of the proposed numerical method and approach.

Xiaobing Feng, Cody Lorton

In this paper we propose and analyze an interior penalty discontinuous Galerkin (IP-DG) method using piecewise linear polynomials for the elastic Helmholtz equations with the first order absorbing boundary condition. It is proved that the sesquilinear form for the problem satisfies a generalized weak coercivity property, which immediately infers a stability estimate for the solution of the differential problem in all frequency regimes. It is also proved that the proposed IP-DG method is unconditionally stable with respect to both frequency $ω$ and mesh size $h$. Sub-optimal order (with respect to $h$) error estimates in the broken $H^1$-norm and in the $L^2$-norm are obtained in all mesh regimes. These estimate improve to optimal order when the mesh size $h$ is restricted to the pre-asymptotic regime (i.e., $ω^βh =O(1)$ for some $1\leq β<2$). The novelties of the proposed IP-DG method include: first, the method penalizes not only the jumps of the function values across the element edges but also the jumps of the normal derivatives; second, the penalty parameters are taken as complex numbers with positive imaginary parts. In order to establish the desired unconditional stability estimate for the numerical solution, the main idea is to exploit a (simple) property of linear functions to overcome the main difficulty caused by non-Hermitian nature and strong indefiniteness of the Helmholtz-type problem. The error estimate is then derived using a nonstandard technique adapted from \cite{Feng_Wu09}. Numerical experiments are also presented to validate the theoretical results and to numerically examine the pollution effect (with respect to $ω$) in the error bounds.