TY - JOUR T1 - Linear Advection with Ill-Posed Boundary Conditions via $L^1$-Minimization AU - Guermond , Jean-Luc AU - Popov , Bojan JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 39 EP - 47 PY - 2007 DA - 2007/04 SN - 4 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/849.html KW - finite elements, best $L^1$-approximation, viscosity solution, linear transport, ill-posed problem. AB -

It is proven that in dimension one the piecewise linear best $L^1$-approximation to the linear transport equation equipped with a set of ill-posed boundary conditions converges in $W_{loc}^{1,1}$ to the viscosity solution of the equation and the boundary layer associated with the ill-posed boundary condition is always localized in one mesh cell, i.e., the "last" one.