A Reduced-Dimension Extrapolating Method of Finite Element Solution Coefficient Vectors for Fractional Tricomi-Type Equation

Abstract

We here employ a proper orthogonal decomposition (POD) to reduce the dimensionality of unknown coefficient vectors of finite element (FE) solutions for the fractional Tricomi-type equation and develop a reduced-dimension extrapolating FE (RDEFE) method for the fractional Tricomi-type equation. For this purpose, we first develop an FE method for the fractional Tricomi-type equation and provide the existence, unconditional stability, and error analysis for the FE solutions. We then develop the RDEFE method for the fractional Tricomi-type equation by means of the POD technique and analyze the existence, unconditional stability, and errors for the RDEFE solutions by using the matrix analysis. Lastly, we provide two numerical examples to verify our theoretical results and to explain the advantages of the RDEFE method.