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Double greedy algorithms: Reduced basis methods for transport dominated problems

Wolfgang Dahmen, Christian Plesken, Gerrit Welper (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov n-widths of the solution sets. The central ingredient is the construction of computationally feasible “tight” surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated...

Double weighted commutators theorem for pseudo-differential operators with smooth symbols

Yu-long Deng, Zhi-tian Chen, Shun-chao Long (2021)

Czechoslovak Mathematical Journal

Let - ( n + 1 ) < m - ( n + 1 ) ( 1 - ρ ) and let T a ρ , δ m be pseudo-differential operators with symbols a ( x , ξ ) n × n , where 0 < ρ 1 , 0 δ < 1 and δ ρ . Let μ , λ be weights in Muckenhoupt classes A p , ν = ( μ λ - 1 ) 1 / p for some 1 < p < . We establish a two-weight inequality for commutators generated by pseudo-differential operators T a with weighted BMO functions b BMO ν , namely, the commutator [ b , T a ] is bounded from L p ( μ ) into L p ( λ ) . Furthermore, the range of m can be extended to the whole m - ( n + 1 ) ( 1 - ρ ) .

Doubling properties and unique continuation at the boundary for elliptic operators with singular magnetic fields

Xiangxing Tao (2002)

Studia Mathematica

Let u be a solution to a second order elliptic equation with singular magnetic fields, vanishing continuously on an open subset Γ of the boundary of a Lipschitz domain. An elementary proof of the doubling property for u² over balls centered at some points near Γ is presented. Moreover, we get the unique continuation at the boundary of Dini domains for elliptic operators.

Dual finite element analysis for elliptic problems with obstacles on the boundary. I

Ivan Hlaváček (1977)

Aplikace matematiky

For an elliptic model problem with non-homogeneous unilateral boundary conditions, two dual variational formulations are presented and justified on the basis of a saddle point theorem. Using piecewise linear finite element models on the triangulation of the given domain, dual numerical procedures are proposed. By means of one-sided approximations, some a priori error estimates are proved, assuming that the solution is sufficiently smooth. A posteriori error estimates and two-sided bounds for the...

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