# FEM applied to heat equation and incompatible conditions

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...

• No, but of class $C^2$. I'd be interested in both cases though.