Laplace and Poisson Equations

S. P. Venkateshan () and Prasanna Swaminathan ()
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S. P. Venkateshan: Indian Institute of Technology Madras, Department of Mechanical Engineering
Prasanna Swaminathan: Universite Sorbonne Paris Nord, Charge de recherche 2

Chapter Chapter 13 in Computational Methods in Engineering, 2023, pp 603-649 from Springer

Abstract: Abstract Laplace and PoissonLaplace equation Poisson equation equations are elliptic partial differential equations and occur in many practical situations such as inviscid flow, heat conduction, mass diffusion and electrostatics. These equations are boundary value problems applied over multi-dimensions. The Poisson equation is given by $$ \underbrace{\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} + \frac{\partial ^2 u}{\partial z^2}}_{\nabla ^2 u} = -q(x,y,z)$$ ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 ⏟ ∇ 2 u = - q ( x , y , z ) $$\nabla ^2$$ ∇ 2 (or Laplacian) is the divergence operator. In the above equation, the variable u could be temperature in which case q represents a heat generation term. When solving an electrostatic problem, u represents electric potential and q represents charge density distribution. If the source term q(x, y, z) is zero, we get the Laplace equation. Such equations can be treated analytically under special conditions, and hence numerical treatment of these equations become important. In this chapter, we shall discuss the solution of these equations using FDM and also briefly explain the application of FEM and FVM.

Date: 2023
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DOI: 10.1007/978-3-031-08226-9_13

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