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Consider the problem $$u_t - \Delta u = f \text{ on } \Omega\times (0,T)\\u=0 \text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) \text{ on } \Omega$$ with $g$ NOT vanishing on the boundary. If I use a FEM discretization in space, what is the best a priori error (considering both space and time) that I can hope for? Under which time discretization? Is there anything else involving finite elements (e.g. space-time FEM) that would improve the situation?

Indeed, all the nice error estimates I came across with, assume compatibilty of initial and boundary condition...

Thank you in advance.

Cross posted also on Mathoverflow: here

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    $\begingroup$ You should mention that you asked almost the same question on MO: mathoverflow.net/questions/421988/… $\endgroup$
    – David Ketcheson
    May 8 at 9:01
  • $\begingroup$ Is the domain polygonal? $\endgroup$
    – Wolfgang Bangerth
    May 9 at 17:16
  • $\begingroup$ No, but of class $C^2$. I'd be interested in both cases though. $\endgroup$
    – brtw
    May 10 at 10:29
  • $\begingroup$ I found out DG methods in time to be some solution, maybe something simpler is available? $\endgroup$
    – brtw
    May 10 at 10:31

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