  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.
