ST-Greedy-POD Reduced Basis Method for Parameterized Partial Differential Equations

Many engineering problems require simulations of parameterized partial differential equations. It takes a lot of time to solve this kind of problem when the discretization scale of the equation is larger and the parameter space is more complex. To improve the solving efficiency of partial differential equations with parameters, this paper proposes an ST-Greedy-POD reduced basis method for parameterized PDEs based on the space-time finite element discretization. Firstly, the parameterized PDE is discretized in both space and time via the space-time finite element method, yielding a parameterized algebraic equation. Then, the parameterized equation is separated into the parameter-dependent part and the parameterindependent part. Secondly, the Greedy algorithm is adopted to select optimal parameters from the parameter sample set according to posterior error, and the space-time reduced basis space is constructed iteratively, to further derive the space-time projection matrix. Finally, the parameterized system separated by parameters is projected onto the space-time reduced basis matrix to obtain the space-time parameterized reduced-order model and the corresponding reduced basis algorithm is presented. The proposed method reduces both the time variable and the space variable simultaneously. The results of two numerical examples verify the feasibility and effectiveness of the proposed method.