The FE (Finite Element) & LP (Linear Programming) method and the related methods for optimization of partial differential equation systems

2005 ◽  
pp. 303-312 ◽  
Author(s):  
Tanehiro Futagami
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Element ◽  
Linear Programming ◽  
Programming Method ◽  
Linear Programming Method ◽  
Partial Differential

Adaptive Finite Element Method for Solving Poisson Partial Differential Equation

2021 ◽  
Vol 4 (1) ◽  
pp. 1-18
Author(s):  
Gokul KC ◽  
Ram Prasad Dulal
Keyword(s):  
Differential Equation ◽  
Finite Element Method ◽  
Partial Differential Equation ◽  
Finite Element ◽  
Poisson Equation ◽  
Rectangular Domain ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Partial Differential ◽  
Element Method

Poisson equation is an elliptic partial differential equation, a generalization of Laplace equation. Finite element method is a widely used method for numerically solving partial differential equations. Adaptive finite element method distributes more mesh nodes around the area where singularity of the solution happens. In this paper, Poisson equation is solved using finite element method in a rectangular domain with Dirichlet and Neumann boundary conditions. Posteriori error analysis is used to mark the refinement area of the mesh. Dorfler adaptive algorithm is used to refine the marked mesh. The obtained results are compared with exact solutions and displayed graphically.

scholarly journals A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 349
Author(s):  
Joël Chaskalovic
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Finite Element ◽  
Finite Elements ◽  
A Priori Estimates ◽  
Probabilistic Approach ◽  
Approximation Error ◽  
Random Variables ◽  
Partial Differential

A probabilistic approach is developed for the exact solution u to a deterministic partial differential equation as well as for its associated approximation uh(k) performed by Pk Lagrange finite element. Two limitations motivated our approach: On the one hand, the inability to determine the exact solution u relative to a given partial differential equation (which initially motivates one to approximating it) and, on the other hand, the existence of uncertainties associated with the numerical approximation uh(k). We, thus, fill this knowledge gap by considering the exact solution u together with its corresponding approximation uh(k) as random variables. By a method of consequence, any function where u and uh(k) are involved are modeled as random variables as well. In this paper, we focus our analysis on a variational formulation defined on Wm,p Sobolev spaces and the corresponding a priori estimates of the exact solution u and its approximation uh(k) in order to consider their respective Wm,p-norm as a random variable, as well as the Wm,p approximation error with regards to Pk finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements Pk1 and Pk2,(k1<k2).

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