Abstract
In this article, a proper orthogonal decomposition (POD) technique is employed to establish a POD-based reduced-order time-space continuous finite element (TSCFE) extrapolation iterative format for two-dimensional (2D) heat equations, which includes very few degrees of freedom but holds sufficiently high accuracy. The error estimates of the POD-based reduced-order TSCFE solutions and the algorithm implementation of the POD-based reduced-order TSCFE extrapolation iterative format are provided. A numerical example is used to illustrate that the results of the numerical computation are consistent with the theoretical conclusions. Moreover, it is shown that the POD-based reduced-order TSCFE extrapolation iterative format is feasible and efficient for solving 2D heat equations.
Highlights
The time-space finite element (FE) methods for time-dependent partial differential equations (TDPDEs) play an important role in many practical applications and form an important research topic
Even if the time-space continuous finite element (TSCFE) methods for two-dimensional ( D) heat equations include a lot of degrees of freedom too, they would cause many difficulties for real-life applications
The proper orthogonal decomposition (POD) method is an efficient means to lessen the degrees of freedom of numerical models for TDPDEs and alleviate the accumulation of truncation errors in the computational process so as to reduce the computational load and save memory requirements
Summary
The time-space finite element (FE) methods for time-dependent partial differential equations (TDPDEs) play an important role in many practical applications and form an important research topic (see [ – ]). If the source term f (x, y, t), the initial value function φ (x, y), the time step k, and the spatial mesh size h all are given, we can obtain solutions uhk(x, y, t) by solving Problem III or the system of equations Based on Sdm( ), the POD-based reduced-order TSCFE extrapolation iterative format for the D heat equations is established as follows. ) from Problem III taking φ = φd, we obtain the following system of error equations: uht k – udt k, φtd + a uhk – udk, φtd dt = , ∀φd ∈ Udnk, where n = M + , M + , . Finding the solutions of the POD-based reduced-order TSCFE extrapolation iterative format for D heat equations consists of the following six steps. Solving Problem IV with d degrees of freedom obtains udk(x, y, t)
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
