Piecewise affine approximations for functions of bounded variation
Provides theoretical guarantees for approximating BV and W^{1,1} functions with piecewise affine functions, benefiting numerical analysis and image processing by improving error estimates from L^1 to W^{1,1} norm.
This work demonstrates that functions of bounded variation (BV) can be well approximated by countably piecewise affine functions, achieving area-strict closeness and arbitrarily small L^1 trace differences on a substantial part of the mesh. For W^{1,1} Sobolev functions, an optimal W^{1,1} error estimate is established for approximation by piecewise affine functions on uniform regular triangulations.
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 Clément (RAIRO Analyse Numérique 9 (1975), no.~R-2, 77--84) and Verfürth (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.