  BV functions cannot be approximated well by piecewise constant functions, but
this work will show that a good approximation is still possible with
(countably) piecewise affine functions. In particular, this approximation is
area-strictly close to the original function and the $\mathrm{L}^1$-difference
between the traces of the original and approximating functions on a substantial
part of the mesh can be made arbitrarily small. Necessarily, the mesh needs to
be adapted to the singularities of the BV function to be approximated, and
consequently, the proof is based on a blow-up argument together with explicit
constructions of the mesh. In the case of $\mathrm{W}^{1,1}$-Sobolev functions
we establish an optimal $\mathrm{W}^{1,1}$-error estimate for approximation by
piecewise affine functions on uniform regular triangulations. The piecewise
affine functions are standard quasi-interpolants obtained by mollification and
Lagrange interpolation on the nodes of triangulations, and the main new
contribution here compared to for instance Cl\'{e}ment (RAIRO Analyse
Num\'{e}rique 9 (1975), no.~R-2, 77--84) and Verf\"{u}rth (M2AN
Math.~Model.~Numer.~Anal. 33 (1999), no. 4, 695-713) is that our error
estimates are in the $\mathrm{W}^{1,1}$-norm rather than merely the
$\mathrm{L}^1$-norm.
