We present a finite element based computational framework to model electrochemical systems. The electrochemical system is represented by the coupled Poisson–Nernst–Planck (PNP) and Navier–Stokes (NS) equations. The key quantity of interest in such simulations is the current (flux) at the system boundaries. Accurately computing the current flux is challenging due to the small critical dimension of the boundary layers (small Debye layer) that require fine mesh resolution at the boundaries. We present a numerical framework which resolves this challenge by utilizing a weak imposition of Dirichlet boundary conditions for Poisson–Nernst–Planck equations. In this numerical framework we utilize a block iterative strategy to solve NS and PNP equations. This allows us to efficiently and easily implement the weak imposition of Dirichlet boundary conditions. The results from our numerical framework shows excellent agreement when compared to strong imposition of boundary conditions (strong imposition requires a much finer mesh). Furthermore, we show that the weak imposition of the boundary conditions allows us to resolve the fluxes in the boundary layers with much coarser meshes compared to strong imposition. We also show that the method converges as we refine the mesh near the boundaries at a much faster rate compared to strong imposition of the boundary layer. We present multiple test cases with varying boundary layer thickness to illustrate the utility of the numerical framework. We illustrate the approach on canonical 3D problems that otherwise would have been computationally intractable to solve accurately. Lastly, we simulate electrokinetic instabilities near a perm-selective membrane with weakly imposed boundary conditions on the membrane. This approach substantially reduces the computational cost of modeling thin boundary layers in electrochemical systems.

我们提出了一个基于有限元的计算框架来模拟电化学系统。电化学系统由耦合的 Poisson-Nernst-Planck (PNP) 和 Navier-Stokes (NS) 方程表示。在这种模拟中，感兴趣的关键量是系统边界处的电流（通量）。由于边界层（小德拜层）的临界尺寸很小，需要在边界处进行精细的网格分辨率，因此准确计算当前通量具有挑战性。我们提出了一个数值框架，通过对 Poisson-Nernst-Planck 方程利用弱 Dirichlet 边界条件来解决这一挑战。在这个数值框架中，我们利用块迭代策略来求解 NS 和 PNP 方程。这使我们能够有效且轻松地实现 Dirichlet 边界条件的弱强加。与强施加边界条件（强施加需要更精细的网格）相比，我们的数值框架的结果显示出极好的一致性。此外，我们表明，与强施加相比，边界条件的弱施加允许我们用更粗糙的网格来解决边界层中的通量。我们还表明，与边界层的强施加相比，当我们以更快的速度细化边界附近的网格时，该方法会收敛。我们提出了多个具有不同边界层厚度的测试用例来说明数值框架的实用性。我们说明了规范 3D 问题的方法，否则这些问题在计算上难以准确解决。最后，我们模拟了选择性渗透膜附近的电动不稳定性，在膜上施加了弱边界条件。这种方法大大降低了模拟电化学系统中薄边界层的计算成本。