Vector Curl and Gradient, Lecture Notes - Mathematics - 1, Study notes for Mathematics. University of Oxford

Mathematics

Description: Finite Element methods, Divergence theorem, strong/classical solutions, weak solutions
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FEM – p.1/16
Finite Element Methods
FEM – p.2/16
Finite Element Methods
Boundary Value Problems
Given a domain Rd, d = 1,2or 3with boundary
−∇ · (ku) = fin
αu +βu
n =gon
(P)
k= 1 Poisson’s equation: ∇ · (u) = 2u
x2+2u
y2+2u
z2
β= 0, (wlog α= 1)Dirichlet boundary conditions,
(P)is the Dirichlet Problem for the Poisson equation
α= 0, (wlog β= 1)Neumann boundary conditions,
(P)is the Neumann Problem for the Poisson equation FEM – p.2/16
Examples:
steady heat conduction:
u=temperature, k=thermal conductivity, f=heating
Neumann boundary conditions insulation,
Dirichlet boundary conditions fixed boundary
temperature
k= 1,u=gravitational potential (−∇u=force),
f=mass distribution
k= 1,u=fluid potential (−∇u=velocity)
k=k1,1k1,2
k1,2k2,2
(or in 3-dimensions) permeability tensor arises in
groundwater flow (flow in porous media):
ku=(Darcy) flow velocity
Always xTkx > 0, x 6= 0 ie. kis positive definite FEM – p.3/16
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