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], [2], [3], [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.