Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Stack Internal
Knowledge at work
Bring the best of human thought and AI automation together at your work.
Explore Stack InternalThe fact that the mesh order is 2 is confusing. It becomes clear (I hope) if you do the same thing with mesh order which is 1 :
It seems that in the case of a mesh order 2, the first 21 elements are the same as the 21 elements of order 1, and the new elements beyond 21 (0.05, 0.15 ...)are a addition to implement the order 2 mesh. I have always observed this kind of construction, though it is not documented.
The fact that the mesh order is 2 is confusing. It becomes clear (I hope) if you do the same thing with mesh order which is 1 :
The fact that the mesh order is 2 is confusing. It becomes clear (I hope) if you do the same thing with mesh order 1 :
It seems that in the case of a mesh order 2, the first 21 elements are the same as the 21 elements of order 1, and the new elements beyond 21 (0.05, 0.15 ...)are a addition to implement the order 2 mesh. I have always observed this kind of construction, though it is not documented.
(sol[[1]])["ElementMesh"] ["Coordinates"] {{0.}, {0.1}, {0.2}, {0.3}, {0.4}, {0.5}, {0.6}, {0.7}, {0.8}, {0.9}, \ {1.}, {1.1}, {1.2}, {1.3}, {1.4}, {1.5}, {1.6}, {1.7}, {1.8}, {1.9}, \ {2.}, {0.05}, {0.15}, {0.25}, {0.35}, {0.45}, {0.55}, {0.65}, {0.75}, \ {0.85}, {0.95}, {1.05}, {1.15}, {1.25}, {1.35}, {1.45}, {1.55}, \ {1.65}, {1.75}, {1.85}, {1.95}} EDIT
Some clarifications
The fact that the mesh order is 2 is confusing. It becomes clear (I hope) if you do the same thing with mesh order which is 1 :
w2 = 6; m = 2; T = 2.0; sol = NDSolveValue[{q'[t] == \[Zeta][t], \[Zeta]'[t] + w2*Sin[q[t]] == 0, DirichletCondition[{q[t] == Pi/3., \[Zeta][t] == 0}, t == 0]}, {q, \[Zeta]}, t \[Element] Line[{{0}, {T}}], Method -> { "FiniteElement", "MeshOptions" -> {"MeshOrder" -> 1} }]; (sol[[1]])["ElementMesh"]["MeshOrder"] (sol[[1]])["ElementMesh"]["MeshElements"] (sol[[1]])["ElementMesh"]["Coordinates"] 1
{NDSolve
FEMLineElement[{{1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 7}, {7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}, {12, 13}, {13, 14}, {14, 15}, {15, 16}, {16, 17}, {17, 18}, {18, 19}, {19, 20}, {20, 21}}]}{{0.}, {0.1}, {0.2}, {0.3}, {0.4}, {0.5}, {0.6}, {0.7}, {0.8}, {0.9}, {1.}, {1.1}, {1.2}, {1.3}, {1.4}, {1.5}, {1.6}, {1.7}, {1.8}, {1.9}, {2.}}
(sol[[1]])["ElementMesh"] ["Coordinates"] {{0.}, {0.1}, {0.2}, {0.3}, {0.4}, {0.5}, {0.6}, {0.7}, {0.8}, {0.9}, \ {1.}, {1.1}, {1.2}, {1.3}, {1.4}, {1.5}, {1.6}, {1.7}, {1.8}, {1.9}, \ {2.}, {0.05}, {0.15}, {0.25}, {0.35}, {0.45}, {0.55}, {0.65}, {0.75}, \ {0.85}, {0.95}, {1.05}, {1.15}, {1.25}, {1.35}, {1.45}, {1.55}, \ {1.65}, {1.75}, {1.85}, {1.95}} (sol[[1]])["ElementMesh"] ["Coordinates"] {{0.}, {0.1}, {0.2}, {0.3}, {0.4}, {0.5}, {0.6}, {0.7}, {0.8}, {0.9}, \ {1.}, {1.1}, {1.2}, {1.3}, {1.4}, {1.5}, {1.6}, {1.7}, {1.8}, {1.9}, \ {2.}, {0.05}, {0.15}, {0.25}, {0.35}, {0.45}, {0.55}, {0.65}, {0.75}, \ {0.85}, {0.95}, {1.05}, {1.15}, {1.25}, {1.35}, {1.45}, {1.55}, \ {1.65}, {1.75}, {1.85}, {1.95}} EDIT
Some clarifications
The fact that the mesh order is 2 is confusing. It becomes clear (I hope) if you do the same thing with mesh order which is 1 :
w2 = 6; m = 2; T = 2.0; sol = NDSolveValue[{q'[t] == \[Zeta][t], \[Zeta]'[t] + w2*Sin[q[t]] == 0, DirichletCondition[{q[t] == Pi/3., \[Zeta][t] == 0}, t == 0]}, {q, \[Zeta]}, t \[Element] Line[{{0}, {T}}], Method -> { "FiniteElement", "MeshOptions" -> {"MeshOrder" -> 1} }]; (sol[[1]])["ElementMesh"]["MeshOrder"] (sol[[1]])["ElementMesh"]["MeshElements"] (sol[[1]])["ElementMesh"]["Coordinates"] 1
{NDSolve
FEMLineElement[{{1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 7}, {7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}, {12, 13}, {13, 14}, {14, 15}, {15, 16}, {16, 17}, {17, 18}, {18, 19}, {19, 20}, {20, 21}}]}{{0.}, {0.1}, {0.2}, {0.3}, {0.4}, {0.5}, {0.6}, {0.7}, {0.8}, {0.9}, {1.}, {1.1}, {1.2}, {1.3}, {1.4}, {1.5}, {1.6}, {1.7}, {1.8}, {1.9}, {2.}}
sol is 2nd order mesh. You can see this with the folllowing code :
(sol[[1]])["ElementMesh"] ["MeshOrder"] 2
In 1D, the elements are segments of lines. Because the order is 2, there are 3 points per segments. The points indices are :
(sol[[1]])["ElementMesh"] ["MeshElements"] {NDSolve
FEMLineElement[{{1, 2, 22}, {2, 3, 23}, {3, 4, 24}, {4, 5, 25}, {5, 6, 26}, {6, 7, 27}, {7, 8, 28}, {8, 9, 29}, {9, 10, 30}, {10, 11, 31}, {11, 12, 32}, {12, 13, 33}, {13, 14, 34}, {14, 15, 35}, {15, 16, 36}, {16, 17, 37}, {17, 18, 38}, {18, 19, 39}, {19, 20, 40}, {20, 21, 41}}]}
The corresponding locations are :
(sol[[1]])["ElementMesh"] ["Coordinates"] {{0.}, {0.1}, {0.2}, {0.3}, {0.4}, {0.5}, {0.6}, {0.7}, {0.8}, {0.9}, \ {1.}, {1.1}, {1.2}, {1.3}, {1.4}, {1.5}, {1.6}, {1.7}, {1.8}, {1.9}, \ {2.}, {0.05}, {0.15}, {0.25}, {0.35}, {0.45}, {0.55}, {0.65}, {0.75}, \ {0.85}, {0.95}, {1.05}, {1.15}, {1.25}, {1.35}, {1.45}, {1.55}, \ {1.65}, {1.75}, {1.85}, {1.95}}