Adaptive domain decomposition methods for advection dominated equations (Q2785701)

The authors are interested in transmission conditions along the artificial boundary generated by decomposition of a solution domain for a linear advection dominated partial differential equation. These conditions are the means whereby two solutions on two halves of a decomposed domain are iteratively brought into consistency and hence converge to a solution over the undecomposed domain. For advection dominated problems, the artificial boundary can be partly ``inflow'' and partly ``outflow,'' and varying the boundary condition from place to place depending on direction of advection is the ``adaptivity'' mentioned in the title. NEWLINENEWLINENEWLINESeveral combinations of transmission conditions are identified in the paper. These include: Dirichlet on inflow and Neumann on outflow portions of the artificial boundary (ADN); Robin on inflow and Neumann on outflow portions (ARN); Dirichlet on inflow and Robin on outflow portions (ADN); and weighted variants of these. They also consider ``damped'' conditions in which the outflow Neumann condition is relaxed by allowing small jumps in the normal derivative of the solution across the artificial boundary in order to speed convergence for very high speed flows. NEWLINENEWLINENEWLINEThe authors present a complete analysis of the continuous one-dimensional case with constant advection coefficient. An exact solution can be found for this case for each iterate. Consequently, convergence can be studied for each of the several choices of transmission conditions. Considerable guidance in choice of method and parameter values can be gained from this discussion. NEWLINENEWLINENEWLINEA two-dimensional case with constant advection parallel to the \(x\)-axis is also considered. Once again, a great deal can be discovered about the solution of the continuous problem through classical methods, making analysis of convergence rates possible. Furthermore, discretization using square finite elements with bilinear shape functions yields discrete solutions whose values are taken directly from the continuous solutions. This discrete case also yields to analysis by classical methods. NEWLINENEWLINENEWLINETurning to the more general two-dimensional case, the authors restrict themselves to the ARN and damped ARN transmission conditions. They show that the iteration is well-defined at each step and converges. They also show that the error arising from damping is comparable to discretization error in very high speed cases.