A posteriori error estimate of the DSD method for first-order hyperbolic equations
The authors establish a posteriori error estimate based on the discontinuous-streamline diffusion (DSD) method for first order hyperbolic equations. The hyperbolic equation has the form: \[ \left\{\begin{matrix} \beta\nabla u+u &=& f(x),\quad x\in\Omega,\\ u(x) &=& g(x),\quad x\in\Gamma_{-}, \end{matrix}\right. \] where \(\Omega\subset\mathbb R^2\) is a bounded domain with polygonal boundary \(\Gamma\), \(\beta=(\beta_1,\beta_2)\) are the given constants, \(\Gamma=\Gamma_{-}\cup\Gamma_{+}:\) \(\Gamma_{-}=\{x\in\Gamma:\beta\cdot n(x)<0\}\), \(n(x)\) is a unit exterior vector of \(\Gamma\) at \(x\), \(\Gamma_+=\{x\in\Gamma:\beta\cdot n(x)\geq 0\}=\Gamma/\Gamma_{-}\), the inflow boundary and the outflow boundary, respectively. A numerical example is given to illustrate the accuracy of this method.
discontinuous-streamline diffusion method
scientific article; zbMATH DE number 1880786