Analysis of the discontinuous Galerkin method for Hamilton-Jacobi equations
The paper is devoted to the investigation of a discontinuous Galerkin finite element method (proposed by Hu and Shu (to appear)) for nonlinear Hamilton-Jacobi equation \[ \frac{{\partial \varphi }}{{\partial t}} + H(\varphi _{x1} ,\dots ,\varphi _{xn}) = 0\quad \text{in }\Omega \times [0,T],\quad \varphi (x,o) = \varphi ^0 (x). \tag{1} \] The main target is to construct the generalized viscosity solution of the boundary value problem. The complexity is the solution may have discontinuous derivatives in the case of complicated geometry of \(\Omega\). The problem (1) is equivalent to a solution of the weakly hyperbolic system \((w_i = \varphi _{xi})\) \[ \frac{{\partial w_i }}{{\partial t}} + \frac{\partial }{{\partial x_i }}H(w) = 0\quad \text{in }\Omega \times [0,T],\quad w_i (x,o) = D_i \varphi ^0 (x). \] The investigating method is based on transition from this boundary value problems to an equivalent variational on a regular triangulation of \(\Omega\) and the representation of the sought approximate solution \(\varphi\) as a polynomial on each triangle. The variational problem is discretizied by using some quadrature rule for the space variables and a Runge-Kutta method for the time variable. At every time step resulting finite-dimensional problem gives on the first stage some discrete approximation of the gradient components \(w_i \) of \(\varphi \) and on the second stage the polynomial approximation of \(\varphi \) is recovered by a least squares procedure. Theoretical results on accuracy and stability properties of the method are proved for certain cases. Numerical examples are presented.
discontinuous Galerkin finite element method
scientific article; zbMATH DE number 1441191