In[]:=
Needs["NDSolve`FEM`"]​​Ω=Rectangle[{0,0},{1,1}];​​eq1=Derivative[1,0][u][x,y]+Derivative[0,1][v][x,y]​​eq2={u[x,y],v[x,y]}.Inactive[Grad][u[x,y],{x,y}]+Inactive[Div][{p[x,y],0},{x,y}]-Pr*Inactive[Div][{Derivative[1,0][u][x,y],Derivative[0,1][u][x,y]},{x,y}]​​eq3={u[x,y],v[x,y]}.Inactive[Grad][v[x,y],{x,y}]+Inactive[Div][{0,p[x,y]},{x,y}]-Pr*Inactive[Div][{Derivative[1,0][v][x,y],Derivative[0,1][v][x,y]},{x,y}]-Pr*Ra*θ[x,y]​​eq4={u[x,y],v[x,y]}.Inactive[Grad][θ[x,y],{x,y}]-Inactive[Div][{Derivative[1,0][θ][x,y],Derivative[0,1][θ][x,y]},{x,y}]​​Pr=0.71;Ra=10^3;​​OP={eq1==0,eq2==0,eq3==0,eq4==0}​​bc={DirichletCondition[θ[x,y]==1,y==0],DirichletCondition[θ[x,y]==0,x==1],DirichletCondition[θ[x,y]==0,x==1],DirichletCondition[{u[x,y]==0,v[x,y]==0},True],DirichletCondition[p[x,y]==0,x==0&&y==0]}​​A=NDSolve[{OP,bc},{u[x,y],v[x,y],p[x,y],θ[x,y]},{x,0,1},{y,0,1},Method->{"FiniteElement","InterpolationOrder"->{u->2,v->2,p->2,θ->2}}]
Out[]=
(0,1)
v
[x,y]+
(1,0)
u
[x,y]
Out[]=
{u[x,y],v[x,y]}.
∇
{x,y}
u[x,y]+
∇
{x,y}
·{p[x,y],0}-0.71
∇
{x,y}
·
(1,0)
u
[x,y],
(0,1)
u
[x,y]
Out[]=
{u[x,y],v[x,y]}.
∇
{x,y}
v[x,y]-710.θ[x,y]+
∇
{x,y}
·{0,p[x,y]}-0.71
∇
{x,y}
·
(1,0)
v
[x,y],
(0,1)
v
[x,y]
Out[]=
{u[x,y],v[x,y]}.
∇
{x,y}
θ[x,y]-
∇
{x,y}
·
(1,0)
θ
[x,y],
(0,1)
θ
[x,y]
Out[]=

(0,1)
v
[x,y]+
(1,0)
u
[x,y]0,{u[x,y],v[x,y]}.
∇
{x,y}
u[x,y]+
∇
{x,y}
·{p[x,y],0}-0.71
∇
{x,y}
·
(1,0)
u
[x,y],
(0,1)
u
[x,y]0,{u[x,y],v[x,y]}.
∇
{x,y}
v[x,y]-710.θ[x,y]+
∇
{x,y}
·{0,p[x,y]}-0.71
∇
{x,y}
·
(1,0)
v
[x,y],
(0,1)
v
[x,y]0,{u[x,y],v[x,y]}.
∇
{x,y}
θ[x,y]-
∇
{x,y}
·
(1,0)
θ
[x,y],
(0,1)
θ
[x,y]0
Out[]=
{DirichletCondition[θ[x,y]1,y0],DirichletCondition[θ[x,y]0,x1],DirichletCondition[θ[x,y]0,x1],DirichletCondition[{u[x,y]0,v[x,y]0},True],DirichletCondition[p[x,y]0,x0&&y0]}
FindRoot
:The method AffineCovariantNewton failed to compute the next step.
Out[]=
u[x,y]InterpolatingFunction
Domain: {{0.,1.},{0.,1.}}
Output: scalar
[x,y],v[x,y]InterpolatingFunction
Domain: {{0.,1.},{0.,1.}}
Output: scalar
[x,y],p[x,y]InterpolatingFunction
Domain: {{0.,1.},{0.,1.}}
Output: scalar
[x,y],θ[x,y]InterpolatingFunction
Domain: {{0.,1.},{0.,1.}}
Output: scalar
[x,y]
In[]:=
StreamPlot[A,{x,y}∈Rectangle[{0,0},{1,1}]]
Out[]=
In[]:=
Length[A]
Out[]=
4
In[]:=
eq=Laplacian[u[x,y],{x,y}]==-(1+y/(x^2+y^2))*u[x,y]+u[x,y]^3;​​dC=DirichletCondition[u[x,y]==1,True];​​NDSolveValue[{eq,dC},u[x,y],{x,y}∈Disk[{0,0},10]]
FindRoot
:The minimal damping factor of
1
10000
has been reached.
NDSolveValue
:PDESolve could not find a solution.
Out[]=
NDSolveValue
(0,2)
u
[x,y]+
(2,0)
u
[x,y]-1-
y
2
x
+
2
y
u[x,y]+
3
u[x,y]
,DirichletCondition[u[x,y]1,True],u[x,y],{x,y}∈Disk[{0,0},10]
In[]:=
eq1//InputForm
Out[]//InputForm=
Derivative[0, 1][v][x, y] + Derivative[1, 0][u][x, y]
In[]:=
bc//InputForm
Out[]//InputForm=
{DirichletCondition[θ[x, y] == 1, y == 0], DirichletCondition[θ[x, y] == 0, x == 1],
DirichletCondition[θ[x, y] == 0, x == 1], DirichletCondition[{u[x, y] == 0, v[x, y] == 0}, True],
DirichletCondition[p[x, y] == 0, x == 0 && y == 0]}