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Vector Curl and Gradient, Lecture Notes - Mathematics - 1

Mathematics

Post: September 9th, 2011
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Finite Element methods, Divergence theorem, strong/classical solutions, weak solutions
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 Ω ⊂ Rd , d = 1, 2 or 3 with boundary ∂Ω − · (k u) ∂u αu + β ∂n = = (P ) k = 1 ⇒ Poisson’s equation: · ( u) = ∂ 2u ∂x2 f g in Ω on ∂Ω + ∂ 2u ∂y 2 + ∂ 2u ∂z 2 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) • k1,1 k1,2 k= k1,2 k2,2 (or in 3-dimensions) permeability tensor arises in groundwater flow (flow in por..

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