A reduced-order extrapolated technique about the unknown coefficient vectors of solutions in the finite element method for hyperbolic type equation

Abstract

This paper is mainly concerned with developing and establishing the reduced-order extrapolated format about the unknown coefficient vectors in numerical solutions to the finite element (FE) method for the hyperbolic type equation. To this end, the functional-form FE format, the existence, stability, and error estimates of the FE solutions, the matrix-form FE format for the hyperbolic type equation are first proposed. Afterwards, a reduced-order extrapolated FE (ROEFE) format is established by means of a proper orthogonal decomposition (POD) technique, and the existence and stability as well as error estimates of the ROEFE solutions are demonstrated by matrix analysis, leading to an elegant theoretical development. Particularly, our work reveals that the ROEFE format possesses the same basis functions and accuracy as the FE method. Finally, some numerical tests are illustrated to computationally confirm the validity and correctness of the ROEFE format.