New coarse grid operators for highly oscillatory coefficient elliptic problems
The authors develop new coarse grid operators for two-dimensional elliptic equations with highly oscillatory coefficients of the form \[ -\nabla\cdot a^{\varepsilon}(x,y)\nabla u_{\varepsilon} = f(x,y),\quad (x,y)\in \Omega =[0,1]\times[0,1] \] subject to the Dirichlet boundary condition \(u_{\varepsilon}|_{\partial\Omega}= 0\). The function \( a^{\varepsilon}(x,y) = a(x/\varepsilon, y/\varepsilon)\) is strictly positive and the small parameter \(\varepsilon\) represents the length of oscillation. New coarse grid operators are constructed directly based on the homogenized differential operators or hierarchically computed from the finest grid. Numerical calculations show that the new coarse grid operators improve the convergence rate of the multigrid method for elliptic equations with oscillatory coefficients. A more realistic problem of heat conduction in a composite structure is also considered.
scientific article; zbMATH DE number 987978