Energy-Minimizing, Symmetric Discretizations for Anisotropic Meshes and Energy Functional Extrapolation

Self-adjoint differential operators often arise from variational calculus on energy functionals. In this case, a direct discretization of the functional induces operator. Following approach, discrete equations are naturally symmetric if is self-adjoint, property that may be lost when using standard difference formulas nonuniform meshes or operator has varying coefficients. Low order finite element systems can derived by approach in systematic way and logically structured they become compact formulas. Extrapolation used then lead to higher oder approximations A rigorous analysis presented for extrapolation combination with nonstandard integration rules elements. likewise applied matrix-free stencils. our applications, both schemes show up quartic convergence.