A space-time certified reduced basis method for Burgers' equation
Author(s)Yano, Masayuki; Patera, Anthony T.; Urban, Karsten
A space-time hp-interpolation-based certified reduced basis method for Burgers' equation
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We present a space-time interpolation-based certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval (0, T] parametrized with respect to the Peclet number. We first introduce a Petrov–Galerkin space-time finite element discretization which enjoys a favorable inf–sup constant that decreases slowly with Peclet number and final time T. We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi–Rappaz–Raviart a posteriori error bounds. We describe computational offline–online decomposition procedures for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf–sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L[superscript 2]-error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T, in marked contrast to the exponentially growing estimate of the classical formulation for high Peclet number cases.
DepartmentMassachusetts Institute of Technology. Department of Mechanical Engineering
Mathematical Models and Methods in Applied Sciences
Yano, Masayuki, Anthony T. Patera, and Karsten Urban. “A Space-Time Hp-Interpolation-Based Certified Reduced Basis Method for Burgers’ Equation.” Math. Models Methods Appl. Sci. 24, no. 09 (August 2014): 1903–1935.