Added docs for BlockWeightedPartitioner

Closes idaholab#13675

framework/doc/content/source/partitioner/BlockWeightedPartitioner.md

@@ -0,0 +1,34 @@
+
+!template load file=stubs/moose_object.md.template name=BlockWeightedPartitioner syntax=/Mesh/Partitioner/BlockWeightedPartitioner
+
+## Overview
+
+Under multiphysics environment, some mesh blocks have more variables and more work load than others .  In parallel, the work
+is spread out based on partitioners that assign equal number of elements to each processor. This will causes imbalanced simulations.
+BlockWeightedPartitioner allow users to specify different weights for different blocks, e.g., low weights for light blocks
+and high weights for heavy blocks.  Usage:
+
+```
+[Mesh]
+  type = FileMesh
+  file = block_weighted_partitioner.e
+
+  [Partitioner]
+    type = BlockWeightedPartitioner
+    block = '1 2 3'
+    weight = '3 1 10'
+  []
+[]
+```
+
+An example:
+
+!row!
+!col! small=12 medium=6 large=3
+!media weightedpartitioner4.png caption=`Weighted partition into 4 subdomains`
+!col-end!
+
+!col! small=12 medium=6 large=3
+!media noweightedpartitioner4.png caption=`Regular partition into 4 subdomains`
+!col-end!
+!row-end!

test/tests/partitioners/block_weighted_partitioner/tests

@@ -1,5 +1,5 @@
 [Tests]
-  design = '/PetscExternalPartitioner.md'
+  design = '/BlockWeightedPartitioner.md'
   issues = '#13675'
 
   [./block_weighted_partitioner]
@@ -10,6 +10,21 @@
     petsc_version = '>=3.10.0'
     min_parallel = 4
     max_parallel = 4
+    # different mode generates different partition
+    mesh_mode = REPLICATED
   [../]
-
+
+  [./block_weighted_partitioner_distributed]
+    requirement = 'The system shall support block weighted partitioner with distributed mesh'
+    type = 'Exodiff'
+    input = 'block_weighted_partitioner.i'
+    exodiff = 'block_weighted_partitioner_out_distributed.e'
+    cli_args = 'Outputs/file_base=block_weighted_partitioner_out_distributed'
+    petsc_version = '>=3.10.0'
+    min_parallel = 4
+    max_parallel = 4
+    # different mode generates different partition
+    mesh_mode = DISTRIBUTED
+  [../]
+
 []