Maximum-norms error estimates for high-order finite volume schemes over quadrilateral meshes

In this paper, we perform \(L^\infty \) and \(W^{1,\infty }\) error estimates for a class of bi-k finite volume schemes on a quadrilateral mesh for elliptic equations, where \(k\ge 2\) is arbitrary. We show that the errors of the finite volume solution in both the \(L^\infty \) and \(W^{1,\infty }\) norms converge to zero with optimal orders, provided the solution \(u\in W^{k+2,\infty }\). Our analysis is based mainly on an estimate of the difference between the finite volume and the corresponding finite element bilinear forms, as well as some techniques derived for \(L^\infty \) and \(W^{1,\infty }\) estimates of the finite element method. Our theoretical findings are supported by several numerical examples.