A reduced‐order extrapolating space–time continuous finite element method for the 2D Sobolev equation

Abstract

AbstractThis paper mainly concerns with the order reduction to the coefficient vectors of the classical space–time continuous finite element (STCFE) solutions for a two‐dimensional Sobolev equation. The classical STCFE model is first constructed for the governing equation, and the theoretical results of the existence, stability, and convergence are provided for the STCFE solutions. We then employ a proper orthogonal decomposition to develop a reduced‐order extrapolating STCFE (ROESTCFE) vector model with the lower dimension, and demonstrate the existence, stability, and convergence for the ROESTCFE solutions by the matrix means, resulting in the very concise and flexible theoretical analysis. Lastly, we examine the effectiveness of the developed ROESTCFE model by several numerical tests. It is shown that the ROESTCFE method is computationally very cheap in actual applications.