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Application of the FE-Approach
Fig. 9: FE-grid and solution to elliptic pde
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 34 / 51
Indian lake
India’s largest lake, the WULAR LAKE, is situated in the state of Jammu& Kashmir at co-ordinates 34.3388583,74.5648655.
Fig. 10: India’s largest lake
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 35 / 51
Equation and conditions
We restrict the demonstration to the homogeneous Dirichlet problem forthe Poisson equation
−∆u(x , y) = r(x , y) .
The unknown u is interpreted as streamfunction, the sets
u(x , y) = c
are streamlines of a flow in the water body.The vectors
~v(x , y) = (∂yu(x , y),−∂xu(x , y))T
are the velocities of the mass flow.
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 36 / 51
Domain definition
Fig. 11: Boundary
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 37 / 51
Nodes of numerical grid
Fig. 12: Nodes
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 38 / 51
Domain triangularisation
Fig. 13: Mesh
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 39 / 51
Data structure
Fig. 14: Data structuresK. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 40 / 51
Assembly of the stiffness matrix
Fig. 15: A neighborhoodK. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 41 / 51
Example
Node no. 555 has the direct neighbors 130, 924, 165, 838, 926, 836.We need to consider 6 triangles, all have node 555 and two of the list ascorners.For each we take the stiffnesses of the master-element and multiply by theJacobian (using the area of the triangle, i.e. a determinant).
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 42 / 51
Procedure
find all direct neighbors
map the master (unit) element to world triangles
scale by determinant of transformation matrix
add element stiffnesses to global stiffness matrix
We proceed in an analogous way with the load vector (right-hand side),using the basis of piecewise linear functions to interpolate the values ofr(x , y) at the nodes.
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 43 / 51
Solution by FEM
Fig. 16: FEM-solution
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 44 / 51
Velocity field
Fig. 17: Velocities
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 45 / 51
Coveats
Remark
A realistic hydrodynamical model is much more complex, it involves thedepth profile, in- and out-fluxes, wind forces, Coriolis forces, temperatureand salinity . . .
This requires to solve a 3d problem for a system of PDEs.
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 46 / 51
A 3d problem
Fig. 18: A 3d problem (by ComSol)
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 47 / 51
Further Problems
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 48 / 51
PDEs of interest
telegraph equation
KdV equation
Bernoulli-Euler equation
bi-Laplace equation
Lame equations
Navier-Stokes equations
Stokes equations
Hamilton-Jacobi
Schrodinger’s equation
Maxwell equations
K. Frischmuth (IfM UR) Analysis and Numerics of PDEs Summer 2021 49 / 51