This paper discusses anisotropic mesh adaptation, addressing either a local interpolation error, or the error on a functional, or the norm of the approximation error, the two last options using an adjoint state. This is explained with a Poisson model problem. We focus on metric-based mesh adaptation using a priori errors. Continuous metric-based methods were developed for this purpose. They propose a continuous statement of the mesh optimisation problem, which need to be then discretised and solved numerically. Tensorial metric-based methods produce directly a discrete optimal metric for interpolation error equirepartition. The novelty of the present paper is to extend the tensorial discrete method to addressing (1) L 1 errors and (2) adjoint-based analyses, two functionalities already available with continuous metric. A first interest is to be able to compare tensorial and continuous methods when they are applied to the reduction of approximation errors. Second, an interesting feature of the new formulation is a potentially sharper analysis of the approximation error. Indeed, the resulting optimal metric has a different anisotropic component. The novel formulation is then compared with the continuous formulation for a few test cases involving high-gradient layers and gradient discontinuities.