A multiscale Eulerian–Lagrangian localized adjoint method for transient advection–diffusion equations with oscillatory coefficients

Abstract

We develop a multiscale Eulerian-Lagrangian localized adjoint method for transient linear advectiondiffusion equations with oscillatory coefficients, which arise in mathematical models for describing flow and transport through heterogeneous porous media, composite material design, and other applications.

A probabilistic collocation Eulerian–Lagrangian localized adjoint method on sparse grids for assessingCO2leakage through wells in randomly heterogeneous porous media

“…A feasible objective is to develop a new set of computational methods that can fill in the large gap between these two model types, with a specific focus on more efficient models to solve the complex set of equations, hybrid models that combine both the simplified and the complex in a kind of multi-scale framework, and a more efficient treatment that will allow the probabilistic treatment of the leakage problem to be accomplished much more efficiently. Certain upscaling or multiscale techniques might also need to be applied [63][64][65][66]. (iv) We considered the critical leakage pathways like human-induced, abandoned wells and assumed that they are vertical.…”

A preliminary study on multiscale ELLAM schemes for transient advection‐diffusion equations

“…The multiscale Eulerian-Lagrangian localized adjoint method (MsELLAM) was proposed for resolving transient linear advection-diffusion equations with oscillatory coefficients [10], based on the framework of the MsFEM [8] and the Eulerian-Lagrangian localized adjoint method [11,12]. The use of Eulerian-Lagrangian approach treats advective transport effectively and stabilizes the numerical approximation.…”

Discontinuous Galerkin immersed finite element methods for parabolic interface problems

“…Recently, it starts to gain more attention on developing IFE methods for time-dependent interface problems. For instance, in [32], transient advection-diffusion equations with interfaces was treated by an immersed Eulerian-Lagrangian localized adjoint method. In [24], numerical solution to parabolic interface problem was considered by applying IFE methods together with the Laplacian transform.…”