A Concurrent Global–Local Numerical Method for Multiscale PDEs

Abstract

We present a new hybrid numerical method for multiscale partial differential equations, which simultaneously captures the global macroscopic information and resolves the local microscopic events over regions of relatively small size. The method couples concurrently the microscopic coefficients in the region of interest with the homogenized coefficients elsewhere. The cost of the method is comparable to the heterogeneous multiscale method, while being able to recover microscopic information of the solution. The convergence of the method is proved for problems with bounded and measurable coefficients, while the rate of convergence is established for problems with rapidly oscillating periodic or almost-periodic coefficients. Numerical results are reported to show the efficiency and accuracy of the proposed method.

A Nitsche Hybrid multiscale method with non-matching grids

“…The rate of convergence was derived for the periodic media and the quasi-periodic media. Numerical results in [28] show that this hybrid method is comparable with the classical global-local method from both the accuracy and the efficiency, while it is particularly suitable for the scenario that the microscopic coefficient a ε is only available in part of the domain, while outside this region, the coarse scale information is available for the coefficient fields. The present work is a follow-up of [28], and there are two contributions.…”

“…Numerical results in [28] show that this hybrid method is comparable with the classical global-local method from both the accuracy and the efficiency, while it is particularly suitable for the scenario that the microscopic coefficient a ε is only available in part of the domain, while outside this region, the coarse scale information is available for the coefficient fields. The present work is a follow-up of [28], and there are two contributions. Firstly we employ the variational formulation of Nitsche [37] to solve (1.2), which allows for non-matching meshes across the interface.…”

“…The third method relies on the hybridization idea [28]. One solves the following variational problem: Find v h ∈ X h such that (1.2) b ε ∇v h , ∇w = f, w for all w ∈ X h , where X h is any Lagrange finite element space, and we denote the L 2 (D) inner product by •, • .…”

“…Such numerical interface is caused by the local support of the transition function. The authors in [28] employed a bodyfitted mesh to solve this interface problem (1.2). Highly refined mesh has to be used around the defect region to ensure the conformity of the mesh and the resolution of the local defects.…”

“…We note that Nitsche's method is a powerful tool to deal with the interface problem in finite elements and discontinuous Galerkin method; See, e.g., [9,12,41]. Another contribution is a general method to construct the transition function, which is an essential ingredient of the hybrid method while seems missing in [28], because only the square defects have been dealt with therein, for which the transition function is a tensor product of a spline function in one dimension. It does not seem easy to find such explicit expression of ρ for defects with irregular shape.…”