This paper establishes the shape derivative of geometry-dependent objective functions for use in constrained variational problems. Using a Lagrangian approach, our differentiablity result is based on the theorem of Delfour–Zolésio on directional derivatives with respect to a parameter of shape perturbation. As the key issue of the paper, we analyze the bijection under the kinematic transport of geometries that is needed for function spaces and feasible sets involved in variational problems. Our abstract theoretical result is applied to the Brinkman flow problem under incompressibility and mixed Dirichlet–Neumann boundary conditions, and provides an analytic formula of the shape derivative based on the velocity method.

This paper describes a numerical solution for a two-point boundary value problem. It includes an algorithm for discretization by mixed finite element method. The discrete scheme allows the utilization a finite element method based on piecewise linear approximating functions and we also use the barycentric quadrature rule to compute the stiffness matrix and the L2-norm.

A proper mathematical model given by nonlinear Poisson-Nernst-Planck (PNP) equations which describe electrokinetics of charged species is considered. The model is generalized with entropy variables associating the pressure and quasi-Fermi electrochemical potentials in order to adhere to the law of conservation of mass. Based on a variational principle for suitable free energy, the generalized PNP system is endowed with the structure of a gradient flow. The well-posedness theorems for the mixed formulation (using the entropy variables) of the gradient-flow problem are provided within the Gibbs simplex and supported by a-priori estimates of the solution.

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