A Concurrent Global-local Numerical Method for Multiscale PDEs
This work addresses the challenge of efficiently solving multiscale PDEs while recovering microscopic details, offering a concurrent coupling approach for computational scientists.
The paper introduces a hybrid numerical method for multiscale PDEs that concurrently captures global macroscopic information and resolves local microscopic events, with cost comparable to the heterogeneous multiscale method. Convergence is proved for bounded measurable coefficients, and convergence rates are established for periodic or almost-periodic coefficients.
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.