Abstract
This study is mainly concerned with the reduced-order extrapolating technique about the unknown solution coefficient vectors in the Crank-Nicolson finite element (CNFE) method for the parabolic type partial differential equation (PDE). For this purpose, the CNFE method and the existence, stability, and error estimates about the CNFE solutions for the parabolic type PDE are first derived. Next, a reduced-order extrapolating CNFE (ROECNFE) model in matrix-form is established with a proper orthogonal decomposition (POD) method, and the existence, stability, and error estimates of the ROECNFE solutions are proved by matrix theory, resulting in an graceful theoretical development. Specially, our study exposes that the ROECNFE method has the same basis functions and the same accuracy as the CNFE method. Lastly, some numeric tests are shown to computationally verify the validity and correctness about the ROECNFE method.
Highlights
The finite element (FE) method has been widely used in scientific engineering computations since it was proposed by Turner, Clough, Martin, and Topp in 1956 in order to solve a structural problem
Owing to the FE or Crank-Nicolson finite element (CNFE) method containing a lot of unknowns, the round-off errors in the calculations are rapidly accumulated, resulting in that the gaining numerical solutions emerge with very large deviation and we could not gain the desired numerical solutions
This paper has dealt with the reduced-order of the CNFE method for the parabolic type partial differential equation (PDE) by means of the reduced-order of the unknown coefficient vectors in the CNFE solutions
Summary
The finite element (FE) method has been widely used in scientific engineering computations since it was proposed by Turner, Clough, Martin, and Topp in 1956 in order to solve a structural problem (see [1]). It is established by means of the POD basic vectors formed by the initial few known coefficient vectors of CNFE solutions, which is different from the reduced-order methods based on the continuous POD basic functions It may keep away from the abstract mathematical knowledge such as the functional analysis and optimization theory. As far as we know, there has been no report that the matrix-form ROECNFE model for the parabolic type PDE is established by reducing the order of coefficient vectors of CNFE solutions via the POD basic vecctors, which keeps the same basic functions as the CNFE model. We develop the matrix-form ROECNFE model for the parabolic type PDE based on the POD basic vectors, set up by reducing the order of the solution coefficient vectors in the CNFE method.
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