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We present a simple direct discretization for functionals used in thevariational mesh generation and adaptation. Meshing functionals are discretizedon simplicial meshes and the Jacobian matrix of the continuous coordinatetransformation is approximated by the Jacobian matrices of affine mappingsbetween elements. The advantage of this direct geometric discretization is thatit preserves the basic geometric structure of the continuous functional, whichis useful in preventing strong decoupling or loss of integral constraintssatisfied by the functional. Moreover, the discretized functional is a functionof the coordinates of mesh vertices and its derivatives have a simpleanalytical form, which allows a simple implementation of variational meshgeneration and adaptation on computer. Since the variational mesh adaptation isthe base for a number of adaptive moving mesh and mesh smoothing methods, theresult in this work can be used to develop simple implementations of thosemethods. Numerical examples are given.

A geometric discretization and a simple implementation for variational mesh generation and adaptation