Super-localized orthogonal decomposition for convection-dominated diffusion problems
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This paper presents a multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The application of the solution operator to piecewise constant right-hand sides on some arbitrary coarse mesh defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the L^2-norm, the Galerkin projection onto this generalized finite element space even yields ε-independent error bounds, ε being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a-posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate ε-independent convergence without preasymptotic effects even in the under-resolved regime of large mesh Péclet numbers.
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