A Fast Finite Difference Method for Elliptic PDEs in Domains with Non-Grid Aligned Boundaries with Application to 3D Linear Elasticity Vita Rutka and Andreas Wiegmann Fraunhofer ITWM Kaiserslautern, Germany. rutka@itwm.fhg.de, wiegmann@itwm.fhg.de Summary. The Explicit Jump Immersed Interface Method reduces the irregular domain problem with non-grid aligned boundaries to solving a sequence of problems in a rectangular parallelepiped on a Cartesian grid using standard central finite differences. Each subproblem is solved using a Fast Fourier Transform based fast solver. The resulting method is second order convergent for the displacements in the maximum norm as the grid is refined. It makes the method attractive for applications where information about the local displacements, stresses and strains is needed, like optimal shape design and others. Key words: Elliptic PDE, linear elasticity, irregular domain, finite differ- ences, fast solvers. 1 Model Equations We consider the equations of isotrope linear elasticity (Navier or Lam´ e equa- tions) in the domain Ω ∈ℜ 3 : μΔu +(λ + μ)∇∇ · u = f (λ + μ) . (1) f : Ω →ℜ 3 is the body force and u =(u,v,w) T is the displacement vector. μ and λ are shear and Lam´ e modulus. The stress tensor is σ =   σ xx σ xy σ xz σ xy σ yy σ yz σ xz σ yz σ zz   := μ ( ∇u +(∇u) T ) + λ   ∂ x 0 0 0 ∂ y 0 0 0 ∂ z   u . Boundary conditions are given as prescribed displacements u = u Γ on ∂Ω D or by an acting force σn = g on ∂Ω T , n is the outer normal to Ω (trac- tions). Require ∂Ω D ∪ ∂Ω T = ∂Ω, ∂Ω D ∩ ∂Ω T = ∅ and area(∂Ω D ) ≥ δ> 0.