# Talk by Karol Hajduk

The next seminar “Modelling of materials – theory, model reduction and efficient numerical methods” will take place on Wednesday from 9:00 till 10:00 in the room K3. The talk will be given by Karol Hajduk. Please see the details below.

Speaker: Karol Hajduk
Title: Simultaneous approximation in Lebesgue and Sobolev spaces

Abstract: In this talk, we present a method to approximate functions defined on smooth bounded domains by elements of the eigenspaces of a positive, self-adjoint, compact-inverse operator (e.g. the Laplacian or the Stokes operator) in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces. Due to the construction, our approximations also respect the `side conditions’ that often occur in boundary value problems associated with the particular choice of the operator (for example, Dirichlet boundary data or a divergence-free condition). As a simple application, we prove that all weak solutions of the convective (Darcy)–Brinkman–Forchheimer equations posed on a bounded domain in \$\mathbb{R}^3\$ satisfy the energy equality. This is joint work with Charles Fefferman (Princeton) and James Robinson (Warwick).