arXiv subjects

Wing Tat Leung

Publications and source records attributed to Wing Tat Leung.

At least 19 recordsLinked to original sources

arXiv2026

A Two-Level Preconditioner Based on Dominant Components for Time-Dependent Multiscale High-Contrast Problem

In this work, we develop a two-level overlapping preconditioner for time-dependent problems in high-contrast multiscale media. We present a coarse-space construction based on multiscale methods, with emphasis on the relaxed nonlocal multicontinuum (NLMC) method. The NLMC space can be separated into components representing the high-permeability regions and the low-permeability background. We show that the complete NLMC space effectively preconditions the heterogeneous stiffness operator, whereas the high-permeability component alone captures the contrast-dependent global modes. For suitable small time-step sizes, the mass matrix controls the low-permeability contribution and only the high-permeability component in NLMC space is required for the global coarse correction in the two-level preconditioner. For general time-step sizes, performance can be maintained by using the full NLMC space or augmenting the high-permeability component with a standard multiscale space. The proposed coarse-space construction lowers the computational cost while preserving robustness with respect to coefficient contrast and fine-scale resolution. To further improve efficiency, the multiscale basis functions can be constructed by iteratively solving the relaxed energy-minimizing formulation. We demonstrate the robustness, efficiency and scalability of the proposed method through several numerical experiments.

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arXiv2026

I-MCHM: Interface Multicontinuum Homogenization for Multiscale Elliptic Problems

We introduce an interface multicontinuum homogenization method (I-MCHM) for high-contrast elliptic problems whose subdomains may have distinct microscopic patterns and unequal numbers of continua. Multicontinuum models upscale microscopic fields to macroscopic continuum quantities; here the continua are the high- and low-permeability regions, and the macroscopic variables are the corresponding local averages of the fine-scale solution on observing cells of size $H_ε$. Although the fine-scale solution is continuous across a material interface, these macroscopic continuum fields need not coincide on the interface. Moreover, when adjacent subdomains retain different numbers of continua, a one-to-one coarse transmission condition cannot be defined. Standard subdomain-wise multicontinuum homogenization therefore determines the bulk equations but leaves the interaction among these coarse variables unspecified. I-MCHM closes this gap by constructing two-sided constrained local bases on interface neighborhoods and assembling a coarse bilinear form with bulk cut-cell integrals and interface-segment corrections over the subdomain adjacency graph, without prescribing a pointwise transmission law. Numerical experiments on straight, curved, and triple-junction geometries, including continuum-coupling ablations and unequal continuum counts, demonstrate the accuracy of the resulting upscaled model.

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arXiv2026

An Adaptive and Physics-Preserving Multiscale Method for Two-Phase Flow Simulations in High-Contrast Heterogeneous Porous Media

In this paper, we propose an adaptive physics-preserving multiscale method for incompressible and immiscible two-phase flow in high-contrast porous media. The method couples a physics-preserving implicit-pressure explicit-saturation scheme (P-IMPES) with the mixed constraint energy minimizing generalized multiscale finite element method. The core algorithmic component is an adaptive update strategy for the saturation-dependent coefficient. Since the effective permeability $κ_n=λ_t(S_w^n)K$ depends on the evolving saturation through the total mobility, we introduce an adaptive update algorithm that monitors the variation of the mobility-weighted coefficient and regenerates the multiscale spaces only when a prescribed tolerance is exceeded. A local postprocessing step is further used to recover fine-grid mass conservation. The analysis is a central part of the paper. We prove local conservation for both phases, the unbiased property of the phase formulation, and bounds preservation under a suitable CFL condition. For the advection-dominated case, we establish velocity and saturation error estimates, which clearly identify the contributions from the adaptive tolerance, the coarse mesh size, the spectral approximation, and the front-layer error. Numerical experiments on different high-contrast permeability fields confirm the physical properties of the method and show that smaller adaptive tolerances improve the saturation approximation while avoiding unnecessary updates of the multiscale spaces.

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arXiv2026

Multiscale modeling for problems with high contrast heterogeneous coefficients by the CEM-GMsFEM

This review paper provides a comprehensive overview of the Constrained Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving elliptic PDEs characterized by highly heterogeneous, high-contrast coefficients. We detail the construction of multiscale basis functions via spectral auxiliary spaces, combined with an oversampling strategy that enables localized computations and guarantees exponential error decay. Rigorous error estimates are outlined for reference to confirm the method's optimal convergence and robustness. Numerical simulations are provided to verify the exponential decay property of the multiscale basis functions. Additionally, we discuss and comment several up-to-date applications of CEM-GMsFEMs.

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arXiv2026

Multicontinuum Generalized Multiscale Finite Element Method (MC-GMsFEM). Theory and applications to upscaling of two-phase flow

We develop a multicontinuum Generalized Multiscale Finite Element Method (MC-GMsFEM) for constructing coarse-scale models in heterogeneous media that simultaneously provide accurate numerical approximations and physically consistent macroscopic equations. Classical multiscale methods efficiently approximate fine-scale solutions on coarse grids using localized basis functions, but they do not offer a systematic pathway for deriving macroscopic governing equations. To overcome this limitation, we introduce a unified framework that integrates multiscale finite element constructions with multicontinuum representations. The proposed method builds on the structure of GMsFEM and exploits a representation of multiscale basis functions that separates coarse variables and their gradients. We construct continuum-dependent basis functions using auxiliary fields defined through local problems with integral constraints, ensuring that each basis function is associated with a specific continuum. This leads to a decomposition of the coarse-scale solution into continuum variables and their gradients, establishing a direct connection between multiscale discretizations and multicontinuum homogenization. Compared to existing multicontinuum approaches, the proposed framework provides greater flexibility in handling heterogeneous media with spatially varying numbers of continua and is naturally embedded within a standard finite element setting. This enables both systematic derivation of macroscopic equations and straightforward numerical implementation. We apply the proposed method to the upscaling of two-phase immiscible flow in heterogeneous porous media, where multiple interacting continua, including mobile and trapped phases, arise. With the proposed approaches, we derive new macroscopic models and show that if classical models are used, the errors can be large.

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arXiv2026

Numerical homogenization for indefinite time-harmonic Maxwell equations

We propose a novel numerical homogenization method based on the edge multiscale approach for solving indefinite time-harmonic Maxwell equations in heterogeneous media with large wavenumber. Numerical methods for these equations in homogeneous media with high wavenumber are particularly challenging due to the so-called pollution effect: the mesh size must be significantly smaller than the reciprocal of the wavenumber to achieve a desired accuracy. This challenge is amplified in heterogeneous media, which frequently occur in practical applications such as metamaterial simulations, since resolving the heterogeneity is necessary for obtaining reliable solutions. Our approach overcomes this difficulty by avoiding explicit resolution of the heterogeneity, while employing a mesh size that depends almost linearly on the reciprocal of the wavenumber. The approximation properties and stability of the method rely critically on the development and rigorous analysis of a novel, nonstandard variational formulation, which constitutes the main innovation of this work. Extensive numerical experiments are provided to validate our theoretical findings.

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arXiv2026

Parallel in time partially explicit splitting scheme for high contrast linear multiscale diffusion problems

Solving multiscale diffusion problems is often computationally expensive due to the spatial and temporal discretization challenges arising from high-contrast coefficients. To address this issue, a partially explicit temporal splitting scheme is proposed. By appropriately constructing multiscale spaces, the spatial multiscale property is effectively captured, and it has been demonstrated that the temporal step size is independent of the contrast. To enhance simulation speed, we propose a parallel algorithm for the multiscale flow problem that leverages the partially explicit temporal splitting scheme. The idea is first to evolve the partially explicit system using a coarse time step size, then correct the solution on each coarse time interval with a fine propagator, for which we consider the all-at-once solver. This procedure is then performed iteratively till convergence. We analyze the stability and convergence of the proposed algorithm. The numerical experiments demonstrate that the proposed algorithm achieves high numerical accuracy for high-contrast problems and converges in a relatively small number of iterations. The number of iterations stays stable as the number of coarse intervals increases, thus significantly improving computational efficiency through parallel processing.

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arXiv2025

Dynamic nonlinear multicontinuum homogenization of systems with intrinsically evolving microstructure

In this paper, we propose a multicontinuum homogenization approach for nonlinear problems involving dynamically evolving multiscale media. The main idea of the proposed approach is that one of the fine-scale variables defines continua. It allows us to formulate macroscopic variables and derive new macroscopic models for nonlinear problems, where coefficients can depend on fine-scale functions. As an example, we consider a fingering problem and employ the fine-scale concentration field to define continua. We consider both Galerkin and mixed multicontinuum modeling approaches. In the former, the multicontinuum theory is applied to the pressure and concentration fields; in the latter, it is also applied to the velocity field. In both approaches, we provide multicontinuum expansions, formulate cell problems, and derive the corresponding macroscopic models. We present numerical results for model problems of gravity-driven fingering, viscous fingering, and interface flattening driven by high-contrast flow. The results show that the macroscopic models, derived with the proposed approach, can provide an accurate representation of the coarse-scale solutions.

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arXiv2025

A Randomized GMsFEM with Data-Driven Predictors for Parametric Flow Problems in Multiscale Heterogeneous Media

In this paper, we propose a randomized generalized multiscale finite element method (Randomized GMsFEM) for flow problems with parameterized inputs and high-contrast heterogeneous media. The method employs a data-driven predictor to construct multiscale basis functions in two stages: offline and online. In the offline stage, a snapshot space is generated via spectral decompositions, and a reduced matrix is obtained using SVD to predict eigenfunctions. In the online stage, these eigenfunctions are evaluated for new parameter realizations to construct the multiscale space. Furthermore, our approach addresses the complexity of multiple permeability fields with random inputs and multiple multiscale information, providing accurate and efficient approximations. Moreover, we conduct a rigorous convergence analysis for our Randomized GMsFEM. Finally, we present extensive numerical examples, demonstrating its superior performance compared to the traditional GMsFEM.

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arXiv2025

An exponential integrator multicontinuum homogenization method for fractional diffusion problem with multiscale coefficients

In this paper, we present a robust and fully discretized method for solving the time fractional diffusion equation with high-contrast multiscale coefficients. We establish the homogenized equation using a multicontinuum approach and employ the exponential integrator method for time discretization. The multicontinuum upscaled model captures the physical characteristics of the solution for the high-contrast multiscale problem, including averages and gradient effects in each continuum at the coarse scale. We then use the exponential integration method for the nonlocal time fractional derivative and it can handle semilinear problem in an efficient way. Convergence analysis of the numerical scheme is provided, along with illustrative numerical examples. Our results demonstrate the accuracy, efficiency, and improved stability for varying order of fractional derivatives.

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arXiv2025

Some convergence analysis for multicontinuum homogenization

In this paper, we provide an analysis of a recently proposed multicontinuum homogenization technique. The analysis differs from those used in classical homogenization methods for several reasons. First, the cell problems in multicontinuum homogenization use constraint problems and can not be directly substituted into the differential operator. Secondly, the problem contains high contrast that remains in the homogenized problem. The homogenized problem averages the microstructure while containing the small parameter. In this analysis, we first based on our previous techniques, CEM-GMsFEM, to define a CEM-downscaling operator that maps the multicontinuum quantities to an approximated microscopic solution. Following the regularity assumption of the multicontinuum quantities, we construct a downscaling operator and the homogenized multicontinuum equations using the information of linear approximation of the multicontinuum quantities. The error analysis is given by the residual estimate of the homogenized equations and the well-posedness assumption of the homogenized equations.

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arXiv2025

Multiscale Partially Explicit Splitting with Mass Lumping for High-Contrast Wave Equations

In this paper, contrast-independent partially explicit time discretization for wave equations in heterogeneous high-contrast media via mass lumping is concerned. By employing a mass lumping scheme to diagonalize the mass matrix, the matrix inversion procedures can be avoided, thereby significantly enhancing computational efficiency especially in the explicit part. In addition, after decoupling the resulting system, higher order time discretization techniques can be applied to get better accuracy within the same time step size. Furthermore, the spatial space is divided into two components: contrast-dependent ("fast") and contrast-independent ("slow") subspaces. Using this decomposition, our objective is to introduce an appropriate time splitting method that ensures stability and guarantees contrast-independent discretization under suitable conditions. We analyze the stability and convergence of the proposed algorithm. In particular, we discuss the second order central difference and higher order Runge-Kutta method for a wave equation. Several numerical examples are presented to confirm our theoretical results and to demonstrate that our proposed algorithm achieves high accuracy while reducing computational costs for high-contrast problems.

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arXiv2024

Multicontinuum splitting scheme for multiscale flow problems

In this paper, we propose multicontinuum splitting schemes for multiscale problems, focusing on a parabolic equation with a high-contrast coefficient. Using the framework of multicontinuum homogenization, we introduce spatially smooth macroscopic variables and decompose the multicontinuum solution space into two components to effectively separate the dynamics at different speeds (or the effects of contrast in high-contrast cases). By treating the component containing fast dynamics (or dependent on the contrast) implicitly and the component containing slow dynamics (or independent of the contrast) explicitly, we construct partially explicit time discretization schemes, which can reduce computational cost. The derived stability conditions are contrast-independent, provided the continua are chosen appropriately. Additionally, we discuss possible methods to obtain an optimized decomposition of the solution space, which relaxes the stability conditions while enhancing computational efficiency. A Rayleigh quotient problem in tensor form is formulated, and simplifications are achieved under certain assumptions. Finally, we present numerical results for various coefficient fields and different continua to validate our proposed approach. It can be observed that the multicontinuum splitting schemes enjoy high accuracy and efficiency.

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arXiv2024

Learning-based Multi-continuum Model for Multiscale Flow Problems

Multiscale problems can usually be approximated through numerical homogenization by an equation with some effective parameters that can capture the macroscopic behavior of the original system on the coarse grid to speed up the simulation. However, this approach usually assumes scale separation and that the heterogeneity of the solution can be approximated by the solution average in each coarse block. For complex multiscale problems, the computed single effective properties/continuum might be inadequate. In this paper, we propose a novel learning-based multi-continuum model to enrich the homogenized equation and improve the accuracy of the single continuum model for multiscale problems with some given data. Without loss of generalization, we consider a two-continuum case. The first flow equation keeps the information of the original homogenized equation with an additional interaction term. The second continuum is newly introduced, and the effective permeability in the second flow equation is determined by a neural network. The interaction term between the two continua aligns with that used in the Dual-porosity model but with a learnable coefficient determined by another neural network. The new model with neural network terms is then optimized using trusted data. We discuss both direct back-propagation and the adjoint method for the PDE-constraint optimization problem. Our proposed learning-based multi-continuum model can resolve multiple interacted media within each coarse grid block and describe the mass transfer among them, and it has been demonstrated to significantly improve the simulation results through numerical experiments involving both linear and nonlinear flow equations.

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arXiv2024

Multicontinuum homogenization in perforated domains

In this paper, we develop a general framework for multicontinuum homogenization in perforated domains. The simulations of problems in perforated domains are expensive and, in many applications, coarse-grid macroscopic models are developed. Many previous approaches include homogenization, multiscale finite element methods, and so on. In our paper, we design multicontinuum homogenization based on our recently proposed framework. In this setting, we distinguish different spatial regions in perforations based on their sizes. For example, very thin perforations are considered as one continua, while larger perforations are considered as another continua. By differentiating perforations in this way, we are able to predict flows in each of them more accurately. We present a framework by formulating cell problems for each continuum using appropriate constraints for the solution averages and their gradients. These cell problem solutions are used in a multiscale expansion and in deriving novel macroscopic systems for multicontinuum homogenization. Our proposed approaches are designed for problems without scale separation. We present numerical results for two continuum problems and demonstrate the accuracy of the proposed methods.

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arXiv2024

Partially explicit splitting scheme with explicit-implicit-null method for nonlinear multiscale flow problems

In this work, we present an efficient approach to solve nonlinear high-contrast multiscale diffusion problems. We incorporate the explicit-implicit-null (EIN) method to separate the nonlinear term into a linear term and a damping term, and then utilise the implicit and explicit time marching scheme for the two parts respectively. Due to the multiscale property of the linear part, we further introduce a temporal partially explicit splitting scheme and construct suitable multiscale subspaces to speed up the computation. The approximated solution is splitted into these subspaces associated with different physics. The temporal splitting scheme employs implicit discretization in the subspace with small dimension that representing the high-contrast property and uses explicit discretization for the other subspace. We exploit the stability of the proposed scheme and give the condition for the choice of the linear diffusion coefficient. The convergence of the proposed method is provided. Several numerical tests are performed to show the efficiency and accuracy of the proposed approach.

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arXiv2023

Partially Explicit Generalized Multiscale Method for Poroelasticity Problem

We develop a partially explicit time discretization based on the framework of constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for the problem of linear poroelasticity with high contrast. Firstly, dominant basis functions generated by the CEM-GMsFEM approach are used to capture important degrees of freedom and it is known to give contrast-independent convergence that scales with the mesh size. In typical situation, one has very few degrees of freedom in dominant basis functions. This part is treated implicitly. Secondly, we design and introduce an additional space in the complement space and these degrees are treated explicitly. We also investigate the CFL-type stability restriction for this problem, and the restriction for the time step is contrast independent.

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arXiv2023

Bayesian deep operator learning for homogenized to fine-scale maps for multiscale PDE

We present a new framework for computing fine-scale solutions of multiscale Partial Differential Equations (PDEs) using operator learning tools. Obtaining fine-scale solutions of multiscale PDEs can be challenging, but there are many inexpensive computational methods for obtaining coarse-scale solutions. Additionally, in many real-world applications, fine-scale solutions can only be observed at a limited number of locations. In order to obtain approximations or predictions of fine-scale solutions over general regions of interest, we propose to learn the operator mapping from coarse-scale solutions to fine-scale solutions using a limited number (and possibly noisy) observations of the fine-scale solutions. The approach is to train multi-fidelity homogenization maps using mathematically motivated neural operators. The operator learning framework can efficiently obtain the solution of multiscale PDEs at any arbitrary point, making our proposed framework a mesh-free solver. We verify our results on multiple numerical examples showing that our approach is an efficient mesh-free solver for multiscale PDEs.

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