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A brief introduction to the "Finite Element Method" FEM,
using the Galerkin Method is galerkin.pdf
There are entire books on FEM and this just covers one small view.
Examples demonstrating the above galerkin.pdf
are shown below. The examples are for Degree 1 (linear), Order 1 through 4,
Dimension 1 through 7. Several programming languages are shown for
some of the examples. Results were close to the same in time and
accuracy for various languages.
The first two examples use three methods of integration:
Adaptive, Newton-Coates and Gauss-Legendre.
For these cases Gauss-Legendre was best.
fem_checke_la.c first order, one dimension
fem_checke_la_c.out output with debug print
fem_checke_la.adb first order, one dimension
fem_checke_la_ada.out output with debug print
fem_checke_la.f90 first order, one dimension
fem_checke_la_f90.out output with debug print
fem_check4th_la.c fourth order, one dimension
fem_check4th_la_c.out output with debug print
fem_check4th_la.adb fourth order, one dimension
fem_check4th_la_ada.out output with debug print
fem_check4th_la.f90 fourth order, one dimension
fem_check4th_la_f90.out output with debug print
fem_check22_la.c second order, two dimension
fem_check22_la_c.out output with debug print
fem_check22_la.adb second order, two dimension
fem_check22_la_ada.out output with debug print
fem_check22_la.f90 second order, two dimension
fem_check22_la_f90.out output with debug print
fem_check22_la.java second order, two dimension
fem_check22_la_java.out output with debug print
with a small study of grid size and integration order, solution with exp(g(x,y))
fem_check22e_la.java second order, two dimension
fem_check22e_la_java.out output with debug print
fem_check_abc_la.c second order, two dimension
fem_check_abc_la_c.out output with debug print
abc_la.h problem definition
abc_la.c problem definition
fem_check_abc_la.adb second order, two dimension
fem_check_abc_la_ada.out output with debug print
abc_la.ads problem definition
abc_la.adb problem definition
fem_check_abc_la.f90 second order, two dimension
fem_check_abc_la_f90.out output with debug print
abc_la.f90 problem definition
fem_check33_la.c third order, three dimension
fem_check33_la_c.out output with debug print
fem_check33_la.f90 third order, three dimension
fem_check33_la_f90.out output with debug print
fem_check33_la.adb third order, three dimension
fem_check33_la_ada.out output with debug print
fem_check33_la.java third order, three dimension
fem_check33_la_java.out output with debug print
npx=3 gave error 10^-12, npx=4 gave error 10^-13 little better for bigger npx
fem_check44_la.c fourth order, four dimension
fem_check44_la_c.out output with debug print
fem_check44_la.f90 fourth order, four dimension
fem_check44_la_f90.out output with debug print
fem_check44_la.adb fourth order, four dimension
fem_check44_la_ada.out output with debug print
fem_check44_la.java fourth order, four dimension
fem_check44_la_java.out output with debug print
fem_check47h_la.c fourth order, seven dimension
fem_check47h_la_c.out output with debug print
fem_check47h_la.java fourth order, seven dimension
fem_check47h_la_java.out output with debug print
In getting ready for this lecture, I have tried many versions of FEM.
The "historic" or classic methods used very low order orthogonal
polynomials and a piecewise linear approach. The trade-off I found was
the time to solve of very large systems of equations vs the time for
numerical quadrature for a much smaller system of equations.
Another trade-off was the complexity of coding the special cases
that arise in piecewise linear approximations. There are many
published tables and formulas for the general case using many
geometric shapes. What is generally missing is the tables and
formulas for the cells next to the boundary.
I found that developing a library routine for the various derivatives
of phi and using high order Lagrange polynomials led to minimizing
programming errors. Each library routine must, of course, be
thoroughly tested. Here are my Lagrange phi routines and tests.
laphi.h "C" header file
laphi.c code through 4th derivative
test_laphi.c numeric test
test_laphi_c.out test output
plot_laphi.c plot test
laphi.ads Ada package specification
laphi.adb code through 4th derivative
test_laphi.adb numeric test
test_laphi_ada.out test output
laphi.f90 module through 4th derivative
test_laphi.f90 numeric test
test_laphi_f90.out test output
laphi.java class through 4th derivative
test_laphi.java numeric test
test_laphi_java.out test output
Other files, that are needed by some examples above:
aquade.ads Ada package specification
aquade.adb tailored adaptive quadrature
gaulegf.adb Gauss-Legendre quadrature
simeq.adb solve simultaneous equations
real_arrays.ads Ada package specification
real_arrays.adb types Real, Real_Vector, Real_Matrix
aquad3.h "C" header file
aquad3.c tailored adaptive quadrature
aquad3e.f90 tailored adaptive quadrature
gaulegf.f90 Gauss-Legendre quadrature
simeq.f90 solve simultaneous equations
gaulegf.java Gauss-Legendre quadrature
simeq.java solve simultaneous equations
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