Create a point element boundary mesh:

Create a 1D boundary mesh from an implicit region:

Create a line element boundary mesh with an additional node inside the domain:

Create a second-order boundary element mesh:

Boundary markers are useful to specify NeumannValue on a region boundary and are explained further in the section Markers in the Element Mesh Generation.

Create a rectangle with boundary element markers and one additional node inside the domain:

Create the boundary mesh:

Visualize the boundary mesh and the markers:

Point markers are useful to specify DirichletCondition on a region boundary and are explained further in the section Markers in the Element Mesh Generation.

Create a rectangle with a boundary element and point elements with markers and one additional node inside the domain:

Create the boundary mesh:

Visualize the point elements in the boundary mesh with node numbers in blue and markers in red:

Ensure that the discretized boundary representation is close to the boundary:

A helper function to compute the centroid of the boundary segments:

The largest deviation from the true disk is at the centers of the segments:

With a larger accuracy goal, the error is reduced by using more points:

ToBoundaryMesh automatically attributes markers to groups of contiguous boundary segments by comparing the boundary normal vectors.

Create a numerical region:

Create a boundary mesh from the numerical region:

Inspect the boundary element markers present:

Visualize the contiguous boundaries:

Visualize the boundaries with element marker 2 or 4:

Inspect the point element markers present:

Visualize the point markers:

Use the element markers for specifying boundary conditions in NDSolveValue:

DirichletCondition extracts the point element markers, while NeumannValue and PeriodicBoundaryCondition extract boundary element markers.

Visualize the solution:

Point element markers (red) are deduced from boundary markers (blue). If the node's connected boundary element markers are the same, then the node will have the same marker. On corners, the connected boundary markers typically differ, and an arbitrary choice of the available markers is made:

To control the behavior of which marker gets assigned to a corner node, a "PointMarkerFunction" can be specified. In the following case, all nodes on the left get a 2 as a marker, and on the right get a 4. The remaining markers are deduced from the boundary segment markers:

Switch off the automatic boundary grouping:

The only boundary element marker is the default of 0:

The automatic boundary grouping chooses boundary segments to be contiguous by computing the normals of the boundary segments and then computing the dot product of the neighboring segments. When the dot product is above a threshold , the neighboring segments are considered contiguous.

Create a coarse boundary mesh where boundary elements are grouped if the threshold :

Inspect the union of the boundary element markers:

Create a coarse boundary mesh where boundary elements are grouped if the threshold :

Inspect the union of the boundary element markers:

Construct a 1D boundary mesh with an internal boundary:

The boundary elements include the internal boundary at 1.0 (index 2):

In 1D, the boundary normal for an interior boundary is taken to be 0:

Inspect the union of boundary element markers:

Create a boundary element mesh in 3D from a discretized graphic:

Generate a list of colors for the many boundary groups:

Visualize only the boundaries with element marker 1:

Inspect the boundaries with a different element marker:

The automatic boundary grouping can be useful even for complicated geometries with a large number of groups. Import a gear as an element mesh:

Visualize the boundary mesh and the contiguous boundary groups:

Boundary markers are useful to specify NeumannValue on a region boundary and are explained further in the section Markers in the Element Mesh Generation.

Here is a function that computes integer markers for edges, depending on the coordinates of the edge elements:

Visualize the boundary mesh and the boundary markers:

Inspect the boundary elements:

If "PointElementMarkers" are also computed, those are accessible as the second argument in the function given to "BoundaryMarkerFunction".

The "Continuation" method uses a curve continuation method that can in many cases resolve corners, cusps, and sharp changes quite well:

The "RegionPlot" method is based on improving output from RegionPlot and can sometimes be faster:

The "RegionPlot" method has suboptions to improve the quality of the boundary mesh:

Pass a method option to the BoundaryDiscretizeRegion boundary mesh generator:

Specify a region:

Pass a method option to the BoundaryDiscretizeRegion boundary mesh generator to fine-tune the discretization algorithm:

Visualize the boundary mesh as a mesh region:

The "OpenCascade" boundary mesh generator is well suited for 3D BooleanRegion expressions consisting of Graphics3D primitives.

Specify a region:

Use "OpenCascade" as a boundary mesh generator:

Generate colors for the boundary markers:

Visualize the boundary element mesh with highlighted boundary markers:

Replace the internal boundary mesh generator with a custom mesh generator:

Specify options for a custom mesh generator:

Generally, it is not advisable to have meshes with duplicate coordinates, since the mesh generation process becomes unpredictable. By default, duplicate coordinates are deleted:

When duplicate coordinates are deleted, the incidents of the elements my be re-indexed:

Create a numerical region:

Inspect the boundary mesh:

Specify some additional points to be included in the boundary mesh:

Include the points in the boundary mesh:

Visualize the boundary mesh with the included points:

These additional points are considered to be part of the boundary, but need not be on the actual boundary.

Create the full mesh from the numerical region:

Points added with "IncludePoints" will be addressable with DirichletCondition. Additionally, if the predicate in DirichletCondition is True, then the boundary condition will take effect at the include points.

With "MaxBoundaryCellMeasure"->m, a boundary cell size is chosen:

Specify a boundary cell size and a cell size:

With MaxCellMeasure->m, a cell size is chosen and a boundary cell size is set to be appropriate for good-quality cells in the embedding dimension:

A specific length can be specified using MaxCellMeasure->{"Length"->len}:

A helper function to compute the length of the boundary segments:

This gives the maximum lengths of the segments:

Create a full mesh from the boundary mesh:

ToBoundaryMesh by default generates a first-order boundary mesh:

Inspect the mesh order:

Integrate over the boundary mesh and assess the accuracy of the integration:

To obtain a second-order boundary mesh, the option "MeshOrder" may be used:

Inspect the mesh order:

Integrate over the boundary mesh and assess the accuracy of the integration:

More information about the "MeshOrder" option can be found in this section of the Element Mesh Generation tutorial and this section of the Element Mesh Generation tutorial.

Point markers are useful to specify DirichletCondition on a region boundary and are explained further in the section Markers in the Element Mesh Generation.

Here is a function that computes integer markers for point elements, depending on the coordinates of the edge nodes:

Inspect the boundary elements:

Create a boundary mesh with an internal region:

Create a full mesh from the boundary mesh:

Create a boundary mesh with an internal region and a region hole specification:

Investigate the region hole in the boundary mesh:

Create a full mesh from the boundary mesh: