A first-order system least-squares finite element method for the Poisson-Boltzmann equation

2009 ◽  
pp. NA-NA ◽  
Author(s):  
Stephen D. Bond ◽  
Jehanzeb Hameed Chaudhry ◽  
Eric C. Cyr ◽  
Luke N. Olson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boltzmann Equation ◽  
Least Squares ◽  
Order System ◽  
First Order ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Element Method

scholarly journals A Deep Neural Network based on ResNet for Predicting Solutions of Poisson–Boltzmann Equation

Electronics ◽  
2021 ◽  
Vol 10 (21) ◽  
pp. 2627
Author(s):  
In Kwon ◽  
Gwanghyun Jo ◽  
Kwang-Seong Shin
Keyword(s):  
Neural Network ◽  
Finite Element Method ◽  
Finite Element ◽  
Boltzmann Equation ◽  
Mean Squared Error ◽  
Immersed Finite Element Method ◽  
Immersed Finite Element ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Element Method

The Poisson–Boltzmann equation (PBE) arises in various disciplines including biophysics, electrochemistry, and colloid chemistry, leading to the need for efficient and accurate simulations of PBE. However, most of the finite difference/element methods developed so far are rather complicated to implement. In this study, we develop a ResNet-based artificial neural network (ANN) to predict solutions of PBE. Our networks are robust with respect to the locations of charges and shapes of solvent–solute interfaces. To generate train and test sets, we have solved PBE using immersed finite element method (IFEM) proposed in (Kwon, I.; Kwak, D. Y. Discontinuous bubble immersed finite element method for Poisson–Boltzmann equation. Communications in Computational Physics 2019, 25, pp. 928–946). Once the proposed ANNs are trained, one can predict solutions of PBE in almost real time by a simple substitution of information of charges/interfaces into the networks. Thus, our algorithms can be used effectively in various biomolecular simulations including ion-channeling simulations and calculations of diffusion-controlled enzyme reaction rate. The performance of the ANN is reported in the result section. The comparison between IFEM-generated solutions and network-generated solutions shows that root mean squared error are below 5·10−7. Additionally, blow-ups of electrostatic potentials near the singular charge region and abrupt decreases near the interfaces are represented in a reasonable way.

scholarly journals Least-Squares Finite Element Method for a Meso-Scale Model of the Spread of COVID-19

Computation ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 18 ◽  
Author(s):  
Fleurianne Bertrand ◽  
Emilie Pirch
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Least Squares ◽  
Scale Model ◽  
Order System ◽  
Time Behavior ◽  
First Order ◽  
Disease Spreading ◽  
Meso Scale ◽  
Element Method

This paper investigates numerical properties of a flux-based finite element method for the discretization of a SEIQRD (susceptible-exposed-infected-quarantined-recovered-deceased) model for the spread of COVID-19. The model is largely based on the SEIRD (susceptible-exposed-infected-recovered-deceased) models developed in recent works, with additional extension by a quarantined compartment of the living population and the resulting first-order system of coupled PDEs is solved by a Least-Squares meso-scale method. We incorporate several data on political measures for the containment of the spread gathered during the course of the year 2020 and develop an indicator that influences the predictions calculated by the method. The numerical experiments conducted show a promising accuracy of predictions of the space-time behavior of the virus compared to the real disease spreading data.

A Least-Squares/Galerkin Finite Element Method for Incompressible Navier-Stokes Equations

2008 ◽  
Author(s):  
Rajeev Kumar ◽  
Brian H. Dennis
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Least Squares ◽  
Basis Functions ◽  
Galerkin Finite Element Method ◽  
Governing Equations ◽  
Galerkin Finite Element ◽  
First Order ◽  
Convection Dominated ◽  
Element Method

The least-squares finite element method (LSFEM), which is based on minimizing the l2-norm of the residual, has many attractive advantages over Galerkin finite element method (GFEM). It is now well established as a proper approach to deal with the convection dominated fluid dynamic equations. The least-squares finite element method has a number of attractive characteristics such as the lack of an inf-sup condition and the resulting symmetric positive system of algebraic equations unlike GFEM. However, the higher continuity requirements for second-order terms in the governing equations force the introduction of additional unknowns through the use of an equivalent first-order system of equations or the use of C1 continuous basis functions. These additional unknowns lead to increased memory and computing time requirements that have prevented the application of LSFEM to large-scale practical problems, such as three-dimensional compressible viscous flows. A simple finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives, thereby avoiding the need for additional unknowns required by pure a LSFEM approach. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the leastsquares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equalorder basis functions for both pressure and velocity. The stability and accuracy of the method are demonstrated with preliminary results of several benchmark problems solved using low-order C0 continuous elements.

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