Space-time adaptive algorithms for parabolic problems

We developed a space and time adaptive method to simulate electroosmosis and mass transport of a sample concentration within a network of microchannels. The space adaptive criteria is based on an error estimator derived using anisotropic interpolation estimates and a post-processing procedure. In order to improve the accuracy of the numerical solution and to reduce even further the computational cost of the numerical simulation, a time adaptive procedure is combined with the one in space. To do so, a time error estimator is derived for a first model problem, the linear heat equation discretized in time with the Crank-Nicolson scheme. The main difficulty is then to obtain an optimal second order error estimator. Applying standard energy techniques with a continuous, piecewise linear approximation in time fail in recovering the optimal order. To restore the appropriate rate of convergence, a continuous piecewise quadratic polynomial function in time is needed. For this purpose, two different quadratic functions are introduced and two different time error estimators are then derived. It turns out that the second error estimator is more efficient than the first one when considering our adaptive algorithm. Thus, using the second quadratic polynomial, an upper bound for the error is derived for a second model problem, the time-dependent convection-diffusion problem discretized in time with the Crank-Nicolson scheme. The corresponding space and time error estimators are finally used for the numerical simulation of mass transport of a sample concentration within a complex network of microchannels driven by an electroosmotic flow and/or by a pressure-driven flow. Numerical results presented show the efficiency and the robustness of this approach.