Piecewise affine approximations for functions of bounded variation

TL;DR

This paper demonstrates that functions of bounded variation can be effectively approximated by piecewise affine functions with area-strict convergence, providing optimal error estimates in the W^{1,1} norm.

Contribution

It introduces a novel approximation method for BV functions using piecewise affine functions with explicit mesh adaptation and establishes optimal W^{1,1} error estimates.

Findings

Approximation by piecewise affine functions is area-strictly close to BV functions.

Error estimates in the W^{1,1} norm are optimal for Sobolev functions.

Mesh adaptation to singularities is necessary for accurate approximation.

Abstract

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.