<i>Finite-Difference Methods for Partial Differential Equations</i>

“…We start by explaining the applicability of Green's identity to the GLP basis functions. We will then prove that basis functions in (25) are GLP basis functions and discuss the consistency and stability of AES-FEM.…”

Section: Analysis Of Aes-femmentioning

confidence: 99%

“…These generalizations are often collectively referred to as generalized finite difference (or GFD ) methods. The earlier GFD methods were based on polynomial interpolation; see e.g., [25], [12], and [26]. These methods construct a local multivariate polynomial interpolation by requiring the number of points in the stencil to be equal to the number of coefficients in the interpolant.…”

Section: Generalized Finite Difference and Weighted Least Squaresmentioning

confidence: 99%

“…The basis functions in (25) are an example of GLP basis functions. We summarize this key feature in the following theorem.…”

Section: Generalized Lagrange Polynomial Basis Functionsmentioning

confidence: 99%

“…The basis functions based on weighted least squares as defined in (25) are generalized Lagrange polynomial basis functions.…”

Section: Generalized Lagrange Polynomial Basis Functionsmentioning

confidence: 99%

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Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Conley

Delaney

Jiao

2016

Numerical Meth Engineering

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The finite element methods (FEM) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the Adaptive Extended Stencil Finite Element Method (AES-FEM) as a means for overcoming this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial (GLP) basis functions, which we construct using local weighted least-squares approximations. The method preserves the theoretical framework of FEM, and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In addition, AES-FEM can use higher-degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. We describe the formulation and implementation of AES-FEM, and analyze its consistency and stability. We present numerical experiments in both 2D and 3D for the Poisson equation and a time-independent convection-diffusion equation. The numerical results demonstrate that AES-FEM is more accurate than linear FEM, is also more efficient than linear FEM in terms of error versus runtime, and enables much better stability and faster convergence of iterative solvers than linear FEM over poor-quality meshes.

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Interphase mass transfer for laminar concurrent flow of carbon dioxide and water between parallel plates

Tang

Himmelblau

1963

AIChE Journal

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lnterphase moss transfer between carbon dioxide gas and water has been investigated. The physical model used was the laminar concurrent flow of gas and liquid in a wide rectangular channel. Because of the high aspect ratio employed, the model was effectively one of parallel plates of infinite width. The current trend in the study of mass transport involves the adaptation of a mathematical model to a real physical situation by means of the general so-called equations of change. These complex equations usually must be simplified before they can be solved. The results from such a theoretical analysis can then be compared with experimental data to test the validity of the model. This approach, when properly applied, can lead to a clearer understanding of transport processes.

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Finite-Difference Methods for Partial Differential Equations

Cited by 518 publications

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Simulations of thermal tissue coagulation and their value for the planning and monitoring of laser‐induced interstitial thermotherapy (LITT)

Simulations of thermal tissue coagulation and their value for the planning and monitoring of laser‐induced interstitial thermotherapy (LITT)

Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Interphase mass transfer for laminar concurrent flow of carbon dioxide and water between parallel plates

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