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Volume 4, Issue 1
Linear Advection with Ill-Posed Boundary Conditions via $L^1$-Minimization

Jean-Luc Guermond & Bojan Popov

Int. J. Numer. Anal. Mod., 4 (2007), pp. 39-47.

Published online: 2007-04

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  • Abstract

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.

  • Keywords

finite elements, best $L^1$-approximation, viscosity solution, linear transport, ill-posed problem.

  • AMS Subject Headings

65N35, 65N22, 65F05, 35J05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-4-39, author = {Guermond , Jean-Luc and Popov , Bojan}, title = {Linear Advection with Ill-Posed Boundary Conditions via $L^1$-Minimization}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2007}, volume = {4}, number = {1}, pages = {39--47}, abstract = {

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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/849.html} }
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.

Guermond , Jean-Luc and Popov , Bojan. (2007). Linear Advection with Ill-Posed Boundary Conditions via $L^1$-Minimization. International Journal of Numerical Analysis and Modeling. 4 (1). 39-47. doi:
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