Mesh adaptation based on transfinite mean value interpolation Abstract   In this paper, the efficiency in mesh updating (radaptivity) of the Transfinite Mean value Interpolation (TMI) and its generalization (kTMI) are compared on three standardized problems to the wellknown Inverse Distance Weighted interpolation (IDW) and Radial Basis Function interpolation (RBF) for unstructured data points and the new kTransfinite Barycentric Interpolation (kTBI) for structured data points such as, for instance, curves or surfaces in 3D. This is achieved by introducing a dynamical version of these interpolations via an ordinary differential equation that can be solved by standard ODE methods that are more economical than, for instance, solving vector partial differential equations as in the pseudosolid method.  A review of the very recent mathematical foundations of the kTMI and kTBI constructed from the function alone (standard) or from the function and its derivatives (enhanced) is provided in the first part of the paper.

Superconvergent Functional Estimates from TensorProduct Generalized SummationbyParts Discretizations in Curvilinear Coordinates We investigate superconvergent functional estimates in curvilinear coordinates for diagonalnorm tensorproduct generalized summationbyparts operators. We show that interpolationextrapolation operators of degree greater than or equal to 2p are required to preserve at least 2p quadrature accuracy and functional superconvergence in curvilinear coordinates when: (1) the Jacobian of the coordinate transformation is approximated by the same generalized summationbyparts operator that is used to approximate the flux terms and (2) the degree of the generalized summationbyparts operator is lower than the degree of the polynomial used to represent the geometry of interest. Legendrexe2x80x93Gaussxe2x80x93Lobatto and Legendrexe2x80x93Gauss elementtype operators are considered. When the aforementioned condition (2) is violated for the Legendrexe2x80x93Gauss operators, there is an evenxe2x80x93odd quadrature convergence pattern that is explained by the cancellation of the leading truncation error terms for the interpolationextrapolation operators that correspond to the odddegree Legendrexe2x80x93Gauss operators. The theory developed is confirmed through numerical examples with a steady onedimensional problem and the unsteady twodimensional linear convection equation.