Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces

We approximate functions defined on smooth bounded domains by elements of the eigenspaces Laplacian or Stokes operator in such a way that approximations are and converge both Sobolev Lebesgue spaces. prove an abstract result referred to fractional power spaces positive, self-adjoint, compact-inverse operators Hilbert spaces, then obtain our main using explicit form these for Dirichlet operators. As simple application, we all weak solutions convective Brinkman–Forchheimer equations posed domain R 3 ${\mathbb {R}}^3$ satisfy energy equality.