Short Finite Element Implementation Poisson equation with mixed

Index Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Discontinuous Galerkin Method Apr 7, 2011

Motivation Poisson equation with mixed bd cond. : bounded Lipschitz domain with boundary :

Model Problem Model problem : bounded Lipschitz domain with boundary : Dirichlet boundary condition

Model Problem Model problem Weak formulation Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short

Galerkin Discretization Using Standard Galerkin method Find Weak formulation Dong-wook Shin(Dept. of CSE, Yonsei

Galerkin Discretization Using Standard Galerkin method : basis of the finite dimensional space :

Galerkin Discretization Using Standard Galerkin method Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short

Triangulation Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2,

Data representation Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun

Data representation Hat functions Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element

Stiffness matrix Triangular element Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element

Stiffness matrix Triangular element with Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite

Stiffness matrix Quadrilateral element Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element

Assembling RHS Volume forces : center of gravity of : if is triangle :

Incorporating D. C Without incorporating D. C Let Dong-wook Shin(Dept. of CSE, Yonsei Univ.

Result Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2,

Q Reference : Remarks around 50 lines of Matlab : short finite element implementation

Hat function Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun

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Short Finite Element Implementation (Poisson equation with mixed boundary condition) Dong-wook Shin Dept. of CSE, Yonsei Univ. Jun 2, 2011

Index Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Discontinuous Galerkin Method Apr 7, 2011 1 / 18

Motivation Poisson equation with mixed bd cond. : bounded Lipschitz domain with boundary : Dirichlet boundary : Neumann boundary Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 2 / 18

Model Problem Model problem : bounded Lipschitz domain with boundary : Dirichlet boundary condition : Neumann boundary condition Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 3 / 18

Model Problem Model problem Weak formulation Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 4 / 18

Galerkin Discretization Using Standard Galerkin method Find Weak formulation Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 5 / 18

Galerkin Discretization Using Standard Galerkin method : basis of the finite dimensional space : basis of where Weak formulation Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 6 / 18

Galerkin Discretization Using Standard Galerkin method Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 7 / 18

Triangulation Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 8 / 18

Data representation Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 9 / 18

Data representation Hat functions Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 10 / 18

Stiffness matrix Triangular element Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 11 / 18

Stiffness matrix Triangular element with Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 12 / 18

Stiffness matrix Quadrilateral element Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 13 / 18

Stiffness matrix Quadrilateral element Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 14 / 18

Stiffness matrix Quadrilateral element Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 15 / 18

Assembling RHS Volume forces : center of gravity of : if is triangle : if is parallelogram Neumann conditions : center of Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) with length Short Finite Element Implementation Jun 2, 2011 16 / 18

Incorporating D. C Without incorporating D. C Let Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 17 / 18

Result Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011 18 / 18

Q Reference : Remarks around 50 lines of Matlab : short finite element implementation Jochen Alberty, Carstensen and Stefan A. Funken nada 1533@yonsei. ac. kr

Hat function Dong-wook Shin(Dept. of CSE, Yonsei Univ. ) Short Finite Element Implementation Jun 2, 2011