TY - JOUR
T1 - Pure Lagrangian and Semi-Lagrangian Finite Element Methods for the Numerical Solution of Convection-Diffusion Problems
AU - M. Benitez & A. Bermudez
JO - International Journal of Numerical Analysis and Modeling
VL - 2
SP - 271
EP - 287
PY - 2014
DA - 2014/11
SN - 11
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/525.html
KW - convection-diffusion equation, pure Lagrangian method, semi-Lagrangian method, Lagrange-Galerkin method, characteristics method, second order schemes, finite element method.
AB - <p style="text-align: justify;">In this paper we propose a unified formulation to introduce and analyze (pure) Lagrangian
and semi-Lagrangian methods for solving convection-diffusion partial differential equations.
This formulation allows us to state classical and new numerical methods. Several examples
are given. We combine them with finite element methods for spatial discretization. One of the
pure Lagrangian methods we introduce has been analyzed in [4] and [5] where stability and error
estimates for time semi-discretized and fully-discretized schemes have been proved. In this
paper, we prove new stability estimates. More precisely, we obtain an $l^∞(H^1)$ stability estimate
independent of the diffusion coefficient and, if the underlying flow is incompressible, we get a stability
inequality independent of the final time. Finally, the numerical solution of a test problem
is presented that confirms the new stability results.</p>