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mayzal
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- TL;DR Summary
- solving PDE numerically using orthogonal collocation on finite elements method
I’m trying to learn how to apply orthogonal collocation on finite elements method (OCFEM) for PDEs and I’m having a trouble with the number of unknown and equations. Suppose I want to solve a PDE numerically using 2nd order Legendre polynomial in three elements (2 interior collocation points per element). Therefore we use:
##u^1(x,t)=\sum_{i=0}^2 A_i^1(t)P_i^1(x) = A_0^1(t)+A_1^1(t)P_1^1(x)+A_2^1(t)P_2^1(x)##
Where ##P_i^1(x)## is shifted Legendre polynomial of order i to the range of element 1 ##[0 , 1/3]##
##u^2(x,t)=\sum_{i=0}^2 A_i^2(t)P_i^2(x) = A_0^2(t)+A_1^2(t)P_1^2(x)+A_2^2(t)P_2^2(x)##
Where ##P_i^2(x)## is shifted Legendre polynomial of order i to the range of element 2 ##[1/3 , 2/3]##
##u^3(x,t)=\sum_{i=0}^2 A_i^3(t)P_i^3(x) = A_0^3(t)+A_1^3(t)P_1^3(x)+A_2^3(t)P_2^3(x)##
Where ##P_i^2(x)## is shifted Legendre polynomial of order i to the range of element 3 ##[2/3 , 1]##
For example ##u^1(x,t)## will be equal to ##A_0^1(t)+A_1^1(t)(6x-1)+A_2^1(t)(-\frac{1}{2}+\frac{3(6x-1)^2}{2})##
I will be substituting those formulation in the PDE at the interior collocation points but I will be getting more equations than the number of unknowns. The equations:
Unknowns:
In some references, in addition to the continuity equations they are equalizing the derivatives as well which is going to produce even more equations.
Can somebody point out what I'm doing wrong.
Attached is a file describing my attempt at learning this method on one PDE.
##u^1(x,t)=\sum_{i=0}^2 A_i^1(t)P_i^1(x) = A_0^1(t)+A_1^1(t)P_1^1(x)+A_2^1(t)P_2^1(x)##
Where ##P_i^1(x)## is shifted Legendre polynomial of order i to the range of element 1 ##[0 , 1/3]##
##u^2(x,t)=\sum_{i=0}^2 A_i^2(t)P_i^2(x) = A_0^2(t)+A_1^2(t)P_1^2(x)+A_2^2(t)P_2^2(x)##
Where ##P_i^2(x)## is shifted Legendre polynomial of order i to the range of element 2 ##[1/3 , 2/3]##
##u^3(x,t)=\sum_{i=0}^2 A_i^3(t)P_i^3(x) = A_0^3(t)+A_1^3(t)P_1^3(x)+A_2^3(t)P_2^3(x)##
Where ##P_i^2(x)## is shifted Legendre polynomial of order i to the range of element 3 ##[2/3 , 1]##
For example ##u^1(x,t)## will be equal to ##A_0^1(t)+A_1^1(t)(6x-1)+A_2^1(t)(-\frac{1}{2}+\frac{3(6x-1)^2}{2})##
I will be substituting those formulation in the PDE at the interior collocation points but I will be getting more equations than the number of unknowns. The equations:
- 6 equations at interior points (2 in each element)
- 2 continuity equations (between element 1 and 2, in addition to element 2 and 3)
- 2 boundary conditions
- Total: 10 equations
Unknowns:
- ##A_0^1(t), A_1^1(t), A_2^1(t)##
- ##A_0^2(t), A_1^2(t), A_2^2(t)##
- ##A_0^3(t), A_1^3(t), A_2^3(t)##
- Total: 9 unknowns
In some references, in addition to the continuity equations they are equalizing the derivatives as well which is going to produce even more equations.
Can somebody point out what I'm doing wrong.
Attached is a file describing my attempt at learning this method on one PDE.