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We approximate functions defined on smooth bounded domains by elements of theeigenspaces of the Laplacian or the Stokes operator in such a way that theapproximations are bounded and converge in both Sobolev and Lebesgue spaces. Weprove an abstract result referred to fractional power spaces of positive,self-adjoint, compact-inverse operators on Hilbert spaces, and then obtain ourmain result by using the explicit form of these fractional power spaces for theDirichlet Laplacian and Stokes operators. As a simple application, we provethat all weak solutions of the incompressible convective Brinkman--Forchheimerequations posed on a bounded domain in ${\mathbb R}^3$ satisfy the energyequality.

Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces