Consider the following Poisson equation defined on the unit square: â. 2 u. âx. 2. +. â. 2 u. ây. 2. = g (x, y), 0 < x < 1, 0 < y < 1. (1) subject to the following BCs.
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�������������������������������� �������������������� ��������������������� ���������� Consider the following Poisson equation defined on the unit square: ∂2 u ∂x2
+
∂2 u ∂y2
= g (x, y),
0 < x < 1, 0 < y < 1
(1)
subject to the following BCs BC1 :
u (x, 0) = 0
(Dirichlet Boundary Condition)
BC2 :
u (x, 1) = 0
(Dirichlet Boundary Condition)
BC3 :
u (0, y) = 0
(Dirichlet Boundary Condition)
BC4 :
∂u ∂x
(1, y) = 0
(Neumann Boundary Condition)
������������������� In this section we will show that for a certain values of g(x,y) we can find an analytical solution. In particular, we will consider a candidate solution of the form: u(x,y)= α Cos[ n π x] Sin[m π y] - α Sin[m π y] where n and m are integers. Let us test if this solution satisfies our Poisson Equation u[x_, y_] := α Cos[ n π x] Sin[m π y] - α Sin[m π y]
Let us take the Laplacian of this function D[u[x, y], {x, 2}] + D[u[x, y], {y, 2}] // Simplify π2 α m2 - m2 + n2 Cos[n π x] Sin[m π y]
Thus if we define g(x,y) to be g[x_, y_] := π2 α m2 - m2 + n2 Cos[n π x] Sin[m π y]
then our u(x,y) function will satisfy
(2)
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PoissonEquation_FEM.nb
∂2 u ∂x2
+
∂2 u ∂y2
= g (x, y),
0 < x < 1, 0 < y < 1
Let us confirm this that the above expression for g(x,y) will satisfy our Poisson equation D[u[x, y], {x, 2}] + D[u[x, y], {y, 2}] - g[x, y] // Simplify 0
Next let us check if our solution satisfies the BCs At y=0 and y=1 we have Dirichlet BC {u[x, 0], u[x, 1]} {0, - α Sin[m π] + α Cos[n π x] Sin[m π]}
It follows,therefore, that as long as m is an integer the Dirichlet BCs at y=0 and y=1 are satisfied. For example if m=4 {u[x, 0], u[x, 1]} /. m → 4 {0, 0}
Consider next the Dirichlet BC at x=0 for all values of y u[0, y] 0
Thus the Dirichlet BC at x=0 is satisfied. Finally let us test to see if the Neumann condition is satisfied at x=1: D[u[x, y], x] /. x → 1 - n π α Sin[n π] Sin[m π y]
Thus the Neumann condition will be satisfied and long as n is an integer D[u[x, y], x] /. x → 1 /. n → 3 0
������� The following Poisson equation ∂2 u ∂x2
+
∂2 u ∂y2
= π2 α m2 - m2 + n2 Cos[n π x] Sin[m π y],
0 < x < 1, 0 < y < 1
(3)
subject to the following BCs BC1 :
u (x, 0) = 0
(Dirichlet Boundary Condition)
BC2 :
u (x, 1) = 0
(Dirichlet Boundary Condition)
BC3 :
u (0, y) = 0
(Dirichlet Boundary Condition)
(4)
PoissonEquation_FEM.nb
BC4 :
∂u ∂x
(1, y) = 0
(Neumann Boundary Condition)
The solution to this problem is u (x, y) = α Cos[ n π x] Sin[m π y] - α Sin[m π y], such that m, n are integers.
Mathematica��������������� We will take the following parameters for this problem m = 2; n = 1; α = 1; g[x_, y_] := π2 α m2 - m2 + n2 Cos[n π x] Sin[m π y]
This problem can be solved conveniently using Finite Elements sol = NDSolveValue[{Laplacian[u[x, y], {x, y}] - g[x, y] ⩵ NeumannValue[0, x ⩵ 1], DirichletCondition[u[x, 0] ⩵ 0, y == 0], DirichletCondition[u[x, 1] ⩵ 0, y ⩵ 1], DirichletCondition[u[0, y] ⩵ 0, x ⩵ 0]}, u, {x, 0, 1}, {y, 0, 1}] InterpolatingFunction
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Plot3D[sol[x, y], {x, 0, 1}, {y, 0, 1}, AxesLabel → {"y", "x", "u"}, ColorFunction → "TemperatureMap", BoundaryStyle → Thick, Mesh → True, ImageSize → 400]
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PoissonEquation_FEM.nb
������������ ContourPlot[sol[x, y], {x, 0, 1}, {y, 0, 1}, ColorFunction → "TemperatureMap", PlotPoints → 80, PlotLegends → Automatic, FrameLabel → {Style["x", 18], Style["y", 18]}] 1.0
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�������������� Let us now compare the numerical value with the exact solution which is given by u (x, y) = α Cos[ n π x] Sin[m π y] - α Sin[m π y]
We define a function for evaluating the exact solution at any given point in the domain exactSoln[x_, y_] := α Cos[ n π x] Sin[m π y] - α Sin[m π y]
PoissonEquation_FEM.nb
Plot3D[exactSoln[x, y], {x, 0, 1}, {y, 0, 1}, AxesLabel → {"y", "x", "u"}, ColorFunction → "TemperatureMap", BoundaryStyle → Thick, Mesh → True, ImageSize → 400]
ContourPlot[exactSoln[x, y], {x, 0, 1}, {y, 0, 1}, ColorFunction → "TemperatureMap", PlotPoints → 80, PlotLegends → Automatic, FrameLabel → {Style["x", 18], Style["y", 18]}] 1.0
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PoissonEquation_FEM.nb
������������������� To assess the error in our numerical solution we can compute the absolute error at any xi , yj node Abs[ uexaxt (xi , yj ) - unum (xi , yj )] Here is the value of u(x,y) at following x and y values {0.487, 0.7435}: val1 = exactSoln[0.487, 0.7435] 0.958371
The value of our numerical solution at the same nodal values is val2 = sol[0.487, 0.7435] 0.958312
Thus absolute error at the selected nodal point is Abs[val1 - val2] 0.0000585626
The agreement is accurate to 0.006% at the selected nodal value
