Gradient of a vector on a triangulated lattice

Question

Suppose I have a triangulated mesh and I have solved a differential equation problem and have a solution phi. From phi I can construct a vector field, how would I go about computing the gradient of this vector field?

The mesh looks something like this. phi is solved for at the center of all of the cells in the picture.

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I'm not sure how to handle the spatial discretization for a triangulated lattice. Any references would be appreciated.

score 5 · Accepted Answer · answered Dec 23 '14 at 21:13

5

You can use a least squares method to calculate the gradient on unstructured meshes. Here's a presentation on the method. least squares gradient

answered Dec 23 '14 at 21:13

user7257

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Does this method by itself work for computing the gradient of a vector? – wgwz Dec 24 '14 at 14:45
The url doesn't exist anymore. – johnbaltis Jun 18 '18 at 16:55

score 1 · Answer 2 · answered Dec 28 '14 at 19:21

1

The gradient of a vector field is simply a matrix where each row is the gradient of one of the (scalar) components of your vector field. In other words, if you know how to compute the gradient of a scalar solution, you also know how to compute the gradient of a vector field.

Of course, on a triangular mesh with piecewise linear elements, the gradient is then simply a constant on every cell.

answered Dec 28 '14 at 19:21

Wolfgang Bangerth

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What are piecewise linear elements? – wgwz Dec 29 '14 at 16:53
Err, the most elementary elements, with the usual hat functions as shape functions. – Wolfgang Bangerth Dec 29 '14 at 23:32
Here's some stuff on shape functions for those like myself who have never heard of these before. Basic explanation of shape functions.. @WolfgangBangerth I plan to check out Lecture 4 of yours. Thanks for the pointers. – wgwz Dec 30 '14 at 00:06

Gradient of a vector on a triangulated lattice

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