Boundary Homogenization for Periodic Arrays of Absorbers

We introduce a homogenization procedure for reaction-diffusion equations in domains whose boundary consists of small alternating regions with prescribed Dirichlet and Neumann data of comparable areas. The homogenized problem is shown to satisfy an effective Dirichlet boundary condition which depends on the geometry of the small-scale boundary structure. This problem is also related to finding the effective trapping rate for a Brownian particle next to a surface with a periodic array of perfect absorbers. We use the method of optimal geometric grids to numerically solve the unit cell problem of homogenization. The geometric homogenization factor is obtained for a number of cell geometries (stripes, square and hexagonal arrays of disk-shaped absorbers or emitters) as a function of the surface area fraction occupied by the absorbers. Empirical analytical expressions that give excellent fits to data for the entire range of area fractions and correct asymptotic behaviors in the limits of small and large absorber area fractions are proposed.