Optimal error bounds for the two-point flux approximation finite volume scheme

We consider a finite volume scheme with two-point flux approximation (TPFA) to approximate a Laplace problem when the solution exhibits no more regularity than belonging to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 0 Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^1_0(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We define an error between the approximate solution and the exact one, involving their difference and the difference of normal gradients to the faces of the mesh. We prove that this error behaves as the sum of an interpolation error and a conformity error. As a consequence, some error bounds, depending on the regularity of the continuous solution, can be obtained for both the approximate solution and an approximate gradient post-processed from this approximate solution. A numerical example illustrates the error estimate in the context of a solution with minimal regularity. This result is extended to evolution problems discretized via the implicit Euler scheme in an appendix.