Optimal error estimates of the semidiscrete local discontinuous Galerkin methods for high order wave equations (Q2882334)

Discontinuous Galerkin (DG) methods form a particular class of finite element methods for solving numerically partial differential equations (PDEs) that use discontinuous, piecewise polynomials as the solution and test space. Whereas DG schemes were mainly introduced to semidiscretize in space PDEs containing only first order spatial derivatives, semidiscrete local discontinuous Galerkin methods (LDG) constitute a natural extension of DG schemes to partial differential equations containing higher than first order spatial derivatives. To do that, the equation is first rewritten as a first order system and then a discontinuous Galerkin method is applied to the system. In this respect, imposing the appropriate numerical fluxes is of fundamental importance to guarantee stability.NEWLINENEWLINEAlthough LDG methods have been used in practice for solving different classes of partial differential equations, both linear and nonlinear, deriving optimal \(L^2\) estimates when they are applied to high order time-dependent wave equations is by no means an easy task. One of the main difficulties resides in the lack of control on the auxiliary variables in the method (approximating the derivatives of the solution) and the lack of control on the interface boundary terms. If these aspects are not appropriately handled, then \(L^2\) error estimates with suboptimal order can only be achieved (e.g., order \(\mathcal{O}(h^{k+1/2})\) for the LDG method with polynomials of degree at most \(k \geq 0\). On the contrary, in the paper under review, \(L^2\) error estimates of order \(\mathcal{O}(h^{k+2})\) are derived for several classes of high order linear wave equations for the first time. The essential point in the proof consists in obtaining energy stability results for all auxiliary variables and then using special projection techniques to eliminate the jump terms in the equations at the cell boundaries. The technique is thoroughly discussed on three representative examples: the one-dimensional third and fifth order linear wave equations and the multidimensional Schrödinger equation.