Load the package:

Create a uniform mesh for the interval :

Visualize the mesh:

Create a graded mesh with a high coordinate concentration on the left:

Visualize the mesh:

Convert the Line to a graded mesh with a "BothEnds" point distribution using default values:

Visualize the mesh:

Create a 2D graded mesh from a product of two 1D graded meshes:

Create a 1D graded mesh with a "Central" point distribution:

Create a second 1D graded mesh with a "Central" point distribution and 50 elements:

Construct a region product:

Visualize the region product of the anisotropic mesh:

Create a third 1D graded mesh with a "BothEnds" point distribution, 50 elements and an endpoint distance of 1/200:

Create the region product:

Visualize the 3D graded mesh:

Solve the following system of diffusion equations over the domain of :

With initial conditions and and boundary conditions at , and . At , and .

To model the infinite domain, create a left-aligned graded mesh over the Line segments from with 150 elements and a minimal distance of 1/1000:

This mesh allows you to have a fine mesh resolution on the left and only few elements on the right, just to extend the domain far enough to mimic an infinite domain without needing an excessive amount of mesh elements and thus unnecessarily increasing computation time.

Set up the variables and parameters where and :

Set up the PDE with initial conditions :

Solve the PDE:

For comparison, solve the PDE just over the domain :

Visualize the solution in the domain for time :

Note that the extension to a larger domain is necessary to capture the behavior of the boundary condition at infinity.

Over the unit square domain, solve an orthotropic diffusion equation with a large difference in scale of the diffusion coefficient. A constant Dirichlet boundary condition is given on the left and a discontinuous Dirichlet boundary condition is given on the right boundary. Define the PDE:

Define Dirichlet conditions:

Due to the large difference in scale of the diffusion coefficient in the direction, the PDE is essentially one-dimensional in the direction. To deal with the discontinuity of the Dirichlet condition at and , a graded mesh is constructed.

Create a 1D mesh with 10 elements:

Create a graded mesh with "Central" alignment and 50 elements in each:

Construct a region product:

Visualize the region product of the anisotropic mesh:

Solve the PDE:

Visualize the density plot of the result:

Visualize ϕ at x=1:

Visualize the derivative of ϕ in the x direction at x=1:

To assess the quality of the solutions, it is instructive to compare the solution with a graded mesh to a default mesh solution:

Note that the function behavior has less overshoot with the anisotropic mesh:

Note that the anisotropic mesh solution function derivative has more overshoot but in a smaller region.

With "Left" or "Right" alignment, the minimum number of elements is 1:

Use at least one element:

With "Central" or "BothEnds" alignment, the minimum value of , the number of elements, is 2:

Use at least two elements:

A "Right" or "Left" point distribution cannot be created if the minimal distance with the length of the input Line and the element count :

Use a smaller minimal distance :

A "BothEnds" or "Central" point distribution cannot be created if the minimal distance with the length of the input Line and the element count :

Use a smaller minimal distance :

A ToGradedMesh cannot be created if the grading ratio is less than or equal to 0:

Use a positive grading ratio :

Mesh generation will not work as expected when the distance between the closest coordinates is greater than the distance between the endpoints of the input mesh over the number of elements.

Consider an input Line with a distance between endpoints equal to 2 (). The input Line will be converted to a graded mesh with 5 elements (), a distance between closest coordinates equal to 1 () and a "Left" alignment:

Visualize the mesh:

Use a smaller minimal distance:

Visualize the mesh:

The property "GradingRatio" has no meaning for a uniform alignment:

Specify a function to describe a point distribution:

Create a graded mesh where the point distribution is given by a function:

Visualize the mesh:

Compare this to a function that uses Fibonacci numbers as a distribution. Write a function to get a Fibonacci distribution:

Convert the Line to a graded mesh using a function to specify the distribution of points:

Visualize the mesh:

Verify that distance between the elements follows a Fibonacci pattern:

Compare the two different distributions:

Visualize the meshes:

Using a log scale, visualize the node distribution in fibmesh and mesh:

Note that for the construction of the Fibonacci distribution it is not necessary to provide the initial spacing, because a Fibonacci sequence only needs the number of elements, while other distributions use a power law approach to achieve a more uniform grading with a specified minimal distance.