A posteriori error estimates for the coupling equations of scalar conservation laws. (English) Zbl 1190.65138
The paper deals with the following coupling of two conservation laws in one dimension:
\[ u_t+f_R(u)_x-(\varepsilon u)_{xx}=0, \text{} x>0, \; t>0, \]
\[ u_t+f_L(u)_x-(\varepsilon u)_{xx}=0, \text{} x<0, \; t>0, \]
\[ u(x,0)=0, \text{} x \in \mathbb{R}, \]
where \(u:(x,t)\in \mathbb{R} \times \mathbb{R}_+ \to u(x,t)\in \mathbb{R}\) satisfies a suitable “continuity” condition
\[ u(x,t)=u^b(t), \quad t \geq 0 \]
at the interface \(x=0,\) to be compatible with initial condition \(u_0\), \(\varepsilon=\varepsilon(x,t)\) is a positive small viscosity, \(u_0:\; \mathbb{R}\to \mathbb{R}\) is a given function and \(f_\alpha:\; \mathbb{R} \to \mathbb{R}, \; \alpha=L,R,\) denote two smooth functions.
Some a posteriori \(L_2(L_2)\) and \(L_{\infty}(H^{-1})\) residual based error estimates for a finite element method are proposed and a numerical example is given.
\[ u_t+f_R(u)_x-(\varepsilon u)_{xx}=0, \text{} x>0, \; t>0, \]
\[ u_t+f_L(u)_x-(\varepsilon u)_{xx}=0, \text{} x<0, \; t>0, \]
\[ u(x,0)=0, \text{} x \in \mathbb{R}, \]
where \(u:(x,t)\in \mathbb{R} \times \mathbb{R}_+ \to u(x,t)\in \mathbb{R}\) satisfies a suitable “continuity” condition
\[ u(x,t)=u^b(t), \quad t \geq 0 \]
at the interface \(x=0,\) to be compatible with initial condition \(u_0\), \(\varepsilon=\varepsilon(x,t)\) is a positive small viscosity, \(u_0:\; \mathbb{R}\to \mathbb{R}\) is a given function and \(f_\alpha:\; \mathbb{R} \to \mathbb{R}, \; \alpha=L,R,\) denote two smooth functions.
Some a posteriori \(L_2(L_2)\) and \(L_{\infty}(H^{-1})\) residual based error estimates for a finite element method are proposed and a numerical example is given.
Reviewer: Iwan Gawriljuk (Eisenach)
MSC:
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35L65 | Hyperbolic conservation laws |
Keywords:
coupling equations; conservation laws; finite element method; a posteriori error estimates; numerical examplesReferences:
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