In this paper, we analyzed a priori and a posteriori error estimates for the h − p version of the finite element discretization of the elliptic Robin boundary control problem. The conforming $h−p$ finite element method is used. First, we established the optimality conditions for the continuous and discrete optimal control problems, respectively. Then, a priori error estimates of the $h−p$ finite element discretization for the optimal control problem are derived rigorously. Moreover, residual-based a posteriori error estimates are established for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive methods for optimal control problems.

A new class of rational parametrization has been developed and it was used to generate a new family of rational k functions B-splines which depends on an index α ∈ ]−∞ , 0[ ∪ ]1 , +∞[. This family of functions verifies, among other things, the properties of positivity, of partition of the unit and, for a given degree k, constitutes a true basis approximation of continuous functions. We loose, however, the regularity classical optimal linked to the multiplicity of nodes, which we recover in the asymptotic case, when α → ∞. The associated B-splines curves verify the traditional properties particularly that of a convex hull and we see a certain “conjugated symmetry” related to α. The case of open knot vectors without an inner node leads to a new family of rational Bezier curves that will be separately, object of in-depth analysis.

The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the problem defining the interaction between a free fluid and poroelastic structure. The free fluid is governed by the Stokes equations, while the flow in the poroelastic medium is modeled using the Biot poroelasticity system. Equilibrium and kinematic conditions are imposed on the interface. The approach utilizes the same nonconforming Crouzeix-Raviart element discretization on the entire domain [Houédanou Koffi Wilfrid, Results in Applied Mathematics 7 (2020) 100127, Elsevier]. For this discretization, we derive a residual indicator based on the jumps of normal derivative of the nonconforming approximation. Lower and upper bounds form the main results with minimal assumptions on the mesh.