A reduced-order extrapolated natural boundary element method based on POD for the parabolic equation in the 2D unbounded domain

Abstract

In this article, we devote ourselves to the research of order reduction of natural boundary element (NBE) based on a proper orthogonal decomposition (POD) for the parabolic equation in the two-dimensional (2D) unbounded domain. For this purpose, we first build a NBE format for the parabolic equation in the 2D unbounded domain and discuss the existence, stability, and convergence of the NBE solutions. And then, we build a reduced-order NBE extrapolated (RONBEE) format based on POD, analyze the errors between the classical NBE and RON-BEE solutions, and supply the implementation procedure for the RONBEE format. Finally, we utilize some numerical experiments to validate that the numerical computational consequences are consistent with the theoretical ones such that the effectiveness and feasibility of the RONBEE format are further verified. Keywords Natural boundary element • Proper orthogonal decomposition • Parabolic equation in the two-dimensional unbounded domain • Reduced-order extrapolated natural boundary element method • Error estimate Mathematics Subject Classification 65N38 • 65N30 • 65N15 Communicated by Philippe Heluy.

The Crank–Nicolson finite element method for the 2D uniform transmission line equation

“…Provided that it is applied for settling big data processing in artificial intelligence and/or computational linguistics, there will be more than tens of millions unknown numbers. Fortunately, a proper orthogonal decomposition (POD) technique may be used to reduce the unknowns in the CNFE method, which can be used to reduce many numerical methods (see, e.g., [17][18][19][20][21][22]). Our future work is reducing the number of unknowns in the CNFE method by the POD technique.…”

A New Reduced-Dimension Iteration Two-Grid Crank–Nicolson Finite-Element Method for Unsaturated Soil Water Flow Problem

“…Hence, it is necessary to study the TGCNFE method for the unsaturated soil flow equation. A significant amount of numerical simulations (see [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]) have revealed that the POD method plays an important role in lessening the unknowns of numerical models and has been successfully used in various fields, such as fluid mechanics and atmospheric sciences (see [34]), statistics (see [35]), and image recognition and signal processing (see [36]).…”

The Reduced-Order Extrapolating Method about the Crank-Nicolson Finite Element Solution Coefficient Vectors for Parabolic Type Equation

“…It was not until 2001 that the POD method was used to concern with the reduced-order of the Galerkin method for the parabolic type PDEs and to discuss the error estimates of the reduced-order Galerkin solutions ( [11]). The POD method is used to the reduced-order for the classical numerical methods such as the finite difference schemes ( [12][13][14]), the FE methods ( [15][16][17]), the finite volume element (FVE) methods ( [18,19]), the collocation spectral (CS) methods ( [20,21]), and the natural boundary element (NBE) method ( [22,23]). It is also used in the reduced basis methods ( [24,25]).…”

“…Whether the reduced-order Galerkin methods in [11,26,27], the reduced-order FE methods in [15,17], the reduced-order FVE methods in [18,19], the reduced-order CS methods in [20,21], the reduced-order NBE methods in [22,23] and the reduced basis methods in [24,25] are constructed by the continuous POD basic functions. The constructing process of the continuous POD basic functions requires the knowledge of the optimization methods and functional analysis.…”