# Optimized Sponge Layers, Optimized Schwarz Waveform Relaxation Algorithms for Convection-diffusion Problems and Best Approximation

## **Abstract**

When solving an evolution equation in an unbounded domain, various strategies have to be applied, aiming at reducing the number of unknowns and the computational cost, from infinite to a finite and not too large number. Among them are truncation of the domain with a sponge boundary, and Schwarz Waveform Relaxation algorithm with overlap. These problems are closely related, as they both use the Dirichlet-to-Neumann map as a starting point for transparent boundary condition on the one hand, and optimal algorithms on the other hand. Differential boundary conditions can then be obtained by minimization of the reflection coefficients or the convergence rate. In the case of unsteady convection-diffusion problems, this leads to a non standard complex best approximation problem that we present and solve.

## Keywords

Convergence Rate Domain Decomposition Unbounded Domain Absorb Boundary Condition Radiation Boundary Condition## Preview

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