Core Concepts
A hybrid finite element method is proposed that can efficiently solve elliptic problems with heterogeneous and high-contrast coefficients. The method decomposes the solution space into coarse and fine components, and employs a localized spectral decomposition to handle high-contrast coefficients, leading to an accurate and robust numerical scheme.
Abstract
The paper presents a hybrid finite element method for solving elliptic equations with heterogeneous and high-contrast coefficients. The key aspects are:
Primal hybrid formulation: The problem is recast in a weak formulation that depends on a polyhedral mesh, introducing hybrid variables on the element faces.
Space decomposition: The solution space is decomposed into coarse (piecewise constant) and fine components, leading to a system of coupled elliptic problems.
Localized Spectral Decomposition (LSD): To handle high-contrast coefficients, a spectral decomposition of the fine space is performed locally on the element faces. This introduces "slow-decaying modes" that are treated separately, enabling exponential decay of the multiscale basis functions.
Efficient implementation: The method is designed to be computationally efficient, with only local computations required in the pre-processing step. The final system is of size independent of the coefficients.
Theoretical analysis: The well-posedness of the discrete problem is established, and a priori error estimates are derived that mitigate the effect of high-contrast coefficients.
The proposed LSD method is dimensional independent and can be extended to other elliptic problems like elasticity. The key is the use of local eigenvalue problems to enrich the solution space, which makes the method robust with respect to high-contrast coefficients.
Stats
amin|v|^2 ≤ a-(x)|v|^2 ≤ A(x)v·v ≤ a+(x)|v|^2 ≤ amax|v|^2 for all v ∈ R^d
|u - u_h|_H1_A(T_H) ≤ H ‖g‖_L2(Ω)
Quotes
"To make the method robust with respect to high-contrast coefficients, we enrich the space solution via local eigenvalue problems, obtaining optimal a priori error estimate that mitigates the effect of having coefficients with different magnitudes."
"The technique developed is dimensional independent and easy to extend to other elliptic problems such as elasticity."