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

Jing Yang, Zhendong Luo

2020Numerical Methods for Partial Differential Equations22 citationsDOI

Abstract

Abstract This 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.

Topics & Concepts

MathematicsSobolev spaceConvergence (economics)Finite element methodStability (learning theory)Applied mathematicsMathematical analysisDimension (graph theory)Matrix (chemical analysis)Space (punctuation)Reduction (mathematics)GeometryComputer scienceOperating systemEconomicsMaterials scienceThermodynamicsPure mathematicsComposite materialMachine learningEconomic growthPhysicsNumerical methods in engineeringNumerical methods for differential equationsAdvanced Numerical Methods in Computational Mathematics