מושגי ליבה
Developing a novel mixed multiscale method for solving second-order elliptic equations with general L∞-coefficients using the Multiscale Spectral Generalized Finite Element Method.
תקציר
The article introduces a novel multiscale mixed finite element method for solving second-order elliptic equations with general L∞-coefficients arising from flow in highly heterogeneous porous media. The method is based on the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) and focuses on achieving exponential convergence with respect to local degrees of freedom at both continuous and discrete levels. The approach combines local mass conservation properties of mixed finite elements with the advantages of MS-GFEM, presenting a comprehensive framework for addressing challenges in flow simulations due to multiscale coefficient structures. The paper outlines the theoretical foundation, construction of local approximation spaces, and validation through numerical results.
Introduction
- Presents challenges in solving Darcy's law equations due to heterogeneous coefficients.
- Discusses conventional techniques requiring resolution of small-scale features.
- Introduces multiscale methods based on basis functions tailored to encode fine-scale information.
- Highlights preference for locally mass-conservative Darcy velocity using mixed methods.
Development of Mixed MS-GFEM
- Describes pioneering works in developing multiscale methods within different frameworks.
- Outlines key studies focusing on mixed formulations within various methodologies.
- Emphasizes rigorous error estimates for general coefficients as a significant research gap.
Continuous Mixed MS-GFEM
- Formulates second-order elliptic equation in mixed form.
- Defines variational formulation and associated spaces for velocity and pressure fields.
- Details construction of local approximation spaces and coarse trial spaces.
Local Approximations
- Constructs local particular functions by solving Neumann boundary value problems.
- Develops optimal approximation spaces based on singular value decomposition principles.
Inf-sup Stability
- Establishes inf-sup stability by enriching velocity space with local pressure basis functions.
Further Analysis and Validation
Numerical experiments are presented to evaluate the performance of the proposed method.
סטטיסטיקה
Exponential convergence is proven at both continuous and discrete levels.