Application of the FE-Approach

Fig. 9: FE-grid and solution to elliptic pde

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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

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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.

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Domain definition

Fig. 11: Boundary

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Nodes of numerical grid

Fig. 12: Nodes

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Domain triangularisation

Fig. 13: Mesh

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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).

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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.

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Solution by FEM

Fig. 16: FEM-solution

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Velocity field

Fig. 17: Velocities

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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.

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A 3d problem

Fig. 18: A 3d problem (by ComSol)

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Further Problems

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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

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