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    NDSolve`FEM`

    InitializePDECoefficients::femcscd

    Details

    • This message is generated when the PDE is convection dominated, and in some cases numerical instabilities may appear.
    • More information can be found in the Finite Element Usage Tips tutorial.
    • Off[message] switches off the message; On[message] switches it on. For example: Off[InitializePDECoefficients::femcscd].

    Examples

    Basic Examples  (2)

    In this case, the time-dependent PDE is treated as a three-dimensional spatial PDE because an initial condition is missing. The three-dimensional PDE is convection dominated:

    Wolfram Language code: NDSolveValue[{D[T[x, y, t], t] - Laplacian[T[x, y, t], {x, y}] == 0, DirichletCondition[T[x, y, t] == 20, x == 100 && t == 0]}, T[x, y, t], {x, 0, 100}, {y, 0, 100}, {t, 0, 100}]

    The solution is to add an initial condition:

    Wolfram Language code: NDSolveValue[{D[T[x, y, t], t] - Laplacian[T[x, y, t], {x, y}] == 0, DirichletCondition[T[x, y, t] == 20, x == 100], T[x, y, 0] == 20}, T[x, y, t], {x, 0, 100}, {y, 0, 100}, {t, 0, 100}]

    Here is a convection-dominated PDE:

    Wolfram Language code: pde = D[1 / r * D[r * g[r, z], r], r] - g[r, z] + 1 / (2 * Pi * r) == 0

    Solving the PDE with boundary conditions gives a warning:

    Wolfram Language code: if = Interpolation[{{0., 0.}, {1., 0.0634}, {2., 0.0573}, {3., 0.0467}, {4., 0.0378}, {5., 0.0312}, {6., 0.0263}, {7., 0.0227}, {8., 0.0199}, {9., 0.0177}, {10., 0.0159}, {11., 0.0145}, {12., 0.0133}, {13., 0.0122}, {14., 0.0114}, {15., 0.0106}, {16., 0.0099}, {17., 0.0094}, {18., 0.0088}, {19., 0.0084}, {20., 0.008}}]; sol = NDSolveValue[{pde, g[0, z] == if[0], g[20, z] == if[20], g[r, 10] == g[r, -10] == if[r]}, g, {r, 0, 20}, {z, -10, 10}];

    Inspecting the solution reveals a kink in the solution:

    Wolfram Language code: Plot[sol[r, 0], {r, 0, 20}]

    To improve the situation, a finer mesh can be used:

    Wolfram Language code: Needs["NDSolve`FEM`"] mesh = ToElementMesh[Rectangle[{0, -10}, {20, 10}], "MaxBoundaryCellMeasure" -> 0.1, MeshElementType -> TriangleElement]; mesh["Wireframe"]

    Even though the message is still present, the solution looks much better:

    Wolfram Language code: sol2 = NDSolveValue[{pde, g[0, z] == if[0], g[20, z] == if[20], g[r, 10] == g[r, -10] == if[r]}, g, {r, z}∈mesh]; Plot[sol2[r, 0], {r, 0, 20}]