On optimal convergence rate of finite element solutions of boundary value problems on adaptive anisotropic meshes

We describe a new method for generating meshes that minimize the gradient of a discretization error. The key element of this method is construction of a tensor metric from edge-based error estimates. In our papers [1–4] we applied this metric for generating meshes that minimize the gradient of P1-interpolation error and proved that for a mesh with N triangles, the L2-norm of gradient of the interpolation error is proportional to N−1/2. In the present paper we recover the tensor metric using hierarchical a posteriori error estimates. Optimal reduction of the discretization error on a sequence of adaptive meshes will be illustrated numerically for boundary value problems ranging from a linear isotropic diffusion equation to a nonlinear transonic potential equation.