Mengnan Li

Mengnan Li, Eric Chung, Lijian Jiang

In this paper, we present a Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for parabolic equations with multiscale coefficients, arising from applications in porous media. We will present the construction of CEM-GMsFEM and rigorously analyze its convergence for the parabolic equations. The convergence rate is characterized by the coarse grid size and the eigenvalue decay of local spectral problems, but is independent of the scale length and contrast of the media. The analysis shows that the method has a first order convergence rate with respect to coarse grid size in the energy norm and second order convergence rate with respect to coarse grid size in $L^2$ norm under some appropriate assumptions. For the temporal discretization, finite difference techniques are used and the convergence analysis of full discrete scheme is given. Moreover, a posteriori error estimator is derived and analyzed. A few numerical results for porous media applications are presented to confirm the theoretical findings and demonstrate the performance of the approach.

Eric T. Chung, Lijian Jiang, Mengnan Li et al.

Simulating diffusion in heterogeneous media presents a significant computational challenge, as resolving microscopic physical scales traditionally demands excessively fine computational grids. To overcome this barrier, we extend the Localized Subspace Iteration (LSI) framework to multiscale parabolic equations. The proposed method constructs optimal, low-dimensional trial spaces by iteratively approximating the dominant eigenspaces of local inverse operators via Localized Standard Subspace Iteration (LSSI) or Localized Krylov Subspace Iteration (LKSI). Because these LSI basis functions are inherently tailored to capture the slow-decaying, low-frequency modes of the parabolic solution, they naturally suppress error accumulation over long-term integration. To further improve computational efficiency, we decouple the basis construction into an offline phase and implement a contrast-independent, partially explicit temporal splitting scheme for online time-stepping. By explicitly advancing the dominant macroscopic modes while implicitly treating high-frequency microscopic corrections, this scheme guarantees stability without imposing restrictive time-step constraints. We establish rigorous a priori error estimates in both the energy and $L^2$ norms. Numerical experiments illustrate the accuracy and efficiency of the LSI framework, particularly highlighting the LKSI method's advantages in handling high-contrast, complex multiscale media.

Mengnan Li, Jason Miller, Zachary Prince et al.

MOOSEnger is a tool-enabled AI agent tailored to the Multiphysics Object-Oriented Simulation Environment (MOOSE). MOOSE cases are specified in HIT ".i" input files; the large object catalog and strict syntax make initial setup and debugging slow. MOOSEnger offers a conversational workflow that turns natural-language intent into runnable inputs by combining retrieval-augmented generation over curated docs/examples with deterministic, MOOSE-aware parsing, validation, and execution tools. A core-plus-domain architecture separates reusable agent infrastructure (configuration, registries, tool dispatch, retrieval services, persistence, and evaluation) from a MOOSE plugin that adds HIT-based parsing, syntax-preserving ingestion of input files, and domain-specific utilities for input repair and checking. An input precheck pipeline removes hidden formatting artifacts, fixes malformed HIT structure with a bounded grammar-constrained loop, and resolves invalid object types via similarity search over an application syntax registry. Inputs are then validated and optionally smoke-tested with the MOOSE runtime in the loop via an MCP-backed execution backend (with local fallback), translating solver diagnostics into iterative verify-and-correct updates. Built-in evaluation reports RAG metrics (faithfulness, relevancy, context precision/recall) and end-to-end success by actual execution. On a 125-prompt benchmark spanning diffusion, transient heat conduction, solid mechanics, porous flow, and incompressible Navier--Stokes, MOOSEnger achieves a 0.93 execution pass rate versus 0.08 for an LLM-only baseline.