Abstract
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 L2 norm in order to enjoy second-order convergence. Numerical examples for the Poisson problem are computed.
Highlights
We focus on methods which build a somewhat optimal mesh defined by a parametrization using a Riemannian metric
Iso-distribution /equi-repartition Hessian-based methods tend to minimize a Sup or L∞ norm of the interpolation error with respect to a metric considered in a subset of metrics with a prescribed number of vertices
The methods minimizing the Lp norm (p < ∞) of the interpolation error of one or several sensors depending on the CFD solution allows to better capture features of different scales in the solution
Summary
A remarkable feature of the goal-oriented metric-based adaptation of [23, 9] is the complete and coherent mathematical formulation of the mesh adaptation problem It takes the form of the optimization of a well-defined functional, namely the error for a prescribed scalar output, to be minimized with respect to a parameter, the metric, belonging to a well-identified and compact set. Another limitation of a goal-oriented method is the scalar character of the error to reduce It leads to use integrals of solution fields as in (u − uh, g), u being the exact solution, uh its approximation and g a field prescribed by user.
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