FEM – p.1/16

Finite Element Methods

FEM – p.2/16

Finite Element Methods

Boundary Value Problems

Given a domain Ω⊂Rd, d = 1,2or 3with boundary ∂Ω

−∇ · (k∇u) = fin Ω

αu +β∂u

∂n =gon ∂Ω

(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):

−k∇u=(Darcy) ﬂow velocity

Always xTkx > 0, x 6= 0 ie. kis positive deﬁnite FEM – p.3/16

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09/09/2011