A fast direct method of solving hermitian fourth-order finite-element schemes for the poisson equation

1995 ◽  
Vol 74 (6) ◽  
pp. 1371-1376 ◽  
Author(s):  
S. I. Solov’ev
Keyword(s):  
Finite Element ◽  
Poisson Equation ◽  
Fourth Order ◽  
Direct Method ◽  
The Poisson Equation ◽  
Finite Element Schemes ◽  
Fast Direct Method

scholarly journals Multiphysics Simulator for the IPMC Actuator: Mathematical Model, Finite Difference Scheme, Fast Numerical Algorithm, and Verification

Micromachines ◽  
2020 ◽  
Vol 11 (12) ◽  
pp. 1119
Author(s):  
Anton P. Broyko ◽  
Ivan K. Khmelnitskiy ◽  
Eugeny A. Ryndin ◽  
Andrey V. Korlyakov ◽  
Nikolay I. Alekseyev ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Poisson Equation ◽  
Numerical Algorithm ◽  
Direct Method ◽  
Model Parameters ◽  
Water Molecules ◽  
The Poisson Equation ◽  
Oscillation Equation

The article is devoted to the development and creation of a multiphysics simulator that can, on the one hand, simulate the most significant physical processes in the IPMC actuator, and on the other hand, unlike commercial products such as COMSOL, can use computing resources economically. The developed mathematical model is an adjoint differential equation describing the transport of charged particles and water molecules in the ion-exchange membrane, the electrostatic field inside, and the mechanical deformation of the actuator. The distribution of the electrostatic potential in the interelectrode space is located by means of the solution of the Poisson equation with the Dirichlet boundary conditions, where the charge density is a function of the concentration of cations inside the membrane. The cation distribution was obtained by means of the solution of the equation system, in which the fluxes of ions and water molecules are described by the modified Nernst-Planck equations with boundary conditions of the third kind (the Robin problem). The cantilever beam forced oscillation equation in the presence of resistance (allowing for dissipative processes) with assumptions of elasticity theory was used to describe the actuator motion. A combination of the following computational methods was used as a numerical algorithm for the solution: the Poisson equation was solved by a direct method, the modified Nernst-Planck equations were solved by the Newton-Raphson method, and the mechanical oscillation equation was solved using an explicit scheme. For this model, a difference scheme has been created and an algorithm has been described, which can be implemented in any programming language and allows for fast computational experiments. On the basis of the created algorithm and with the help of the obtained experimental data, a program has been created and the verification of the difference scheme and the algorithm has been performed. Model parameters have been determined, and recommendations on the ranges of applicability of the algorithm and the program have been given.

scholarly journals A Note on Multilevel Based Error Estimation

2016 ◽  
Vol 16 (3) ◽  
pp. 447-458 ◽  
Author(s):  
Helmut Harbrecht ◽  
Reinhold Schneider
Keyword(s):  
Finite Element ◽  
Poisson Equation ◽  
Error Estimation ◽  
Finite Element Approximation ◽  
Error Estimator ◽  
Element Approximation ◽  
Finite Element Approximations ◽  
The Poisson Equation ◽  
Galerkin Solution ◽  
Edge Based

AbstractBy employing the infinite multilevel representation of the residual, we derive computable bounds to estimate the distance of finite element approximations to the solution of the Poisson equation. If the finite element approximation is a Galerkin solution, the derived error estimator coincides with the standard element and edge based estimator. If Galerkin orthogonality is not satisfied, then the discrete residual additionally appears in terms of the BPX preconditioner. As a by-product of the present analysis, conditions are derived such that the hierarchical error estimation is reliable and efficient.

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