Finite Element Methods For Elliptic Problems
Uses the Matlab languages for the finite element implementation.
Uses the C library triangle
https://www.cs.cmu.edu/~quake/triangle.html
for 2D mesh generation.
Lab Session 1
- Starting with the triangle library in Matlab.
- Mesh Creation.
- Mesh Acquisition In Matlab.
- Mesh Drawing In Matlab.
- Triangles Count In Plot.
- Vertex Count In Plot.
Lab Session 2
-div(grad u) = f on omega
u = g on omega's border
First Order Elements
- KhTs Matrices Computation.
- General Kh Matrix Computation (Memorized as a sparse matrix).
- fhT Elements With Trapezoid And Barycenter Methods.
- Fh Array Computation.
- Border Condition Implementation.
- Solution Computation.
- Solution Plot.
- Example with f = 1 and g = 0.
Lab Session 3
-div(c grad u) = f on omega
u = g on omega's border
First Order Elements
- Added General Diffusion Term c(x,y)
- c = 1
- c = 2 + x + sin(3y)
- Added Neumann Conditions.
- Discontinuous c(x,y) Example (With Mesh That Follows The Discontinuity).
- Added Exact Solution Confront.
- Added An Order 2 Formula For Numerical Integration.
-div(c grad u) + alpha*u = f on omega
u = g on omega's border
- Added general reaction term alpha.
Lab Session 4
-div(c grad u) = f on omega
Second Order Elements
PART I
- Added mesh details for finite elements of order K = 2. (See http://arxiv.org/abs/math/0501496v2 for quadrature formula)
- Added KE partial coefficients matrix computation.
PART II
- Added Full Kh matrix Computation
- Added FE partial coeficients array computation.
- Added full Fh coeficients array computation.
- Added NonHomogeneous Dirichlet conditions.
- Added uh approximated solution computation.
- Uh plot using only vertices info.
Lab Session 5
-div(c grad u) = f on omega
Second Order Elements
- Plot using edge info.
- Added uniform mesh generator.
- Added exact solution plot.
- L2 error.
- Added makeue.
- Added quadrature formula for degree=3.
Exam Project
-div(c grad u) + beta grad u + alpha u = f on omega
Second Order Elements
- Added transport term: beta grad u.
- Added reaction term: alpha u.