FEniCS Course - FEniCS Project

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Solving PDEs in <strong>FEniCS</strong>Solving a physical problem with <strong>FEniCS</strong> consists of thefollowing steps:1 Identify the PDE and its boundary conditions2 Reformulate the PDE problem as a variational problem3 Make a Python program where the formulas in thevariational problem are coded, along with definitions ofinput data such as f, u 0 , and a mesh for Ω4 Add statements in the program for solving the variationalproblem, computing derived quantities such as ∇u, andvisualizing the results4 / 22
Deriving a variational problem for Poisson’sequationThe simple recipe is: multiply the PDE by a test function v andintegrate over Ω:∫∫− (∆u)v dx = fv dxΩΩThen integrate by parts and set v = 0 on the Dirichletboundary:∫∫∫∂u− (∆u)v dx = ∇u · ∇v dx −ΩΩ∂Ω ∂n v ds} {{ }=0We find that:∫Ω∫∇u · ∇v dx =Ωfv dx5 / 22
Page 1 and 2: FEniCS CourseLecture 2: Static line
Page 3: The FEM cookbookAu = f(i)Partial di
Page 7 and 8: Discrete variational problem for Po
Page 9 and 10: A test problemWe construct a test p
Page 11 and 12: Step by step: the first lineThe fir
Page 13 and 14: Step by step: creating a function s
Page 15 and 16: Step by step: defining a boundary c
Page 17 and 18: Step by step: defining the right-ha
Page 19 and 20: Step by step: solving variational p
Page 21 and 22: Python/FEniCS programming 1011 Open

FEniCS Course - FEniCS Project

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