We present a novel formulation for the mesh adaptation of the approximation of a Partial Differential Equation (PDE). The discussion is restricted to a Poisson problem. The proposed norm-oriented formulation extends the goal-oriented formulation since it is equation-based and uses an adjoint. At the same time, the norm-oriented formulation somewhat supersedes the goal-oriented one since it is basically a solution-convergent method. Indeed, goal-oriented methods rely on the reduction of the error in evaluating a chosen scalar output with the consequence that, as mesh size is increased (more degrees of freedom), only this output is proven to tend to its continuous analog while the solution field itself may not converge. A remarkable quality of goal-oriented metric-based adaptation is the mathematical formulation of the mesh adaptation problem under the form of the optimization, in the well-identified set of metrics, of a well-defined functional. In the new proposed formulation, we amplify this advantage. We search, in the same well-identified set of metrics, the minimum of a norm of the approximation error. The norm is prescribed by the user and the method allows addressing the case of multi-objective adaptation like, for example in aerodynamics, adaptating the mesh for drag, lift and moment in one shot. In this work, we consider the basic linear finite-element approximation and restrict our study to L 2 norm in order to enjoy second-order convergence. Numerical examples for the Poisson problem are computed.

Summary This paper studies the combination of the full‐multigrid (FMG) algorithm with an anisotropic metric‐based mesh adaptation algorithm. For the sake of simplicity, the case of an elliptic two‐dimensional partial differential equation is studied. Meshes are unstructured and non‐embedded, defined through the metric‐based parameterization. A rather classical MG preconditioner is applied, in combination with a quasi‐Newton fixed point. An anisotropic metric‐based mesh adaptation loop is introduced inside the FMG algorithm. FMG convergence stopping test is revisited. Applications to a few two‐dimensional continuous and discontinuous coefficient elliptic model problems show the efficiency of this combination. Copyright © 2015 John Wiley & Sons, Ltd.

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 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.