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A Local-structure-preserving Local Discontinuous Galerkin Method for the Laplace Equation

In this paper, we present a local-structure-preserving local discontinuous Galerkin (LDG) method for the Laplace equation. The method is based on the standard LDG formulation and uses piecewise harmonic polynomials, which satisfy the partial differential equation (PDE) exactly inside each element, as the approximating solutions for the primitive variable u, leading to a significant reduction of the degrees of freedom for the final system and hence the computational cost, without sacrificing the convergence quality of the solutions. An a priori error estimate in the energy norm is established. Numerical experiments are performed to verify optimal convergence rates of the local-structure-preserving LDG method in the energy norm and in the L-norm, as well as to compare it with the standard LDG method to demonstrate comparable performance of the two methods even though the new local-structure-preserving LDG method is computational less expensive.