Chapter 9 The Finite Element Method for 2D elliptic PDEs

IiiiiiChapter 9 The Finite ElementMethod for 2D ellipticPDEsThe procedure of the Finite Element Method to solve 2D problems is the same as that for1D problems, as the flow chart below Integration by parts weak form inV:a(u,v) =L(v)or minv VF(v) Vh( Finite dimensional space and basis functions) a(uh,vh) =L(vh) uhand error The second Green s theorem and integration byparts in 2 DLet us first recall the 2D version of the well known divergencetheorem in Cartesian H1( ) H1( )is a vector in 2D, thenZ Z Fdxdy=Z F nds,( )wherenis the unit normal direction pointing outward at the boundary with line elementds, and is the gradient operator once again is = [ x, y] second Green s theorem is a corollary of the divergence theorem if we setF=v u= v u x,v u y since F= x v u x + y v u y = u x v x+v 2u x2+ u y v y+v 2u y2= u v+v u,219iiiiii220 Chapter 9.

The Finite Element Method for 2D elliptic PDEs The procedure of the finite element method to solve 2D problems is the same as that for 1D problems, as the flow chart below demonstrates. PDE −→ Integration by parts −→ weak form in V: a(u,v) = L(v) or min v∈V F(v) −→ Vh (finite dimensional space and basis functions)

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  Elements, Finite, Finite element

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