diff --git a/dune/multimesh/multigridview.hh b/dune/multimesh/multigridview.hh
index 6f08151ff06ed348988b19c791d9e539fbc6265d..d9942373e36cb0957db72d7fdc0eb6232aac076f 100644
--- a/dune/multimesh/multigridview.hh
+++ b/dune/multimesh/multigridview.hh
@@ -17,46 +17,10 @@
 #include <dune/grid/common/gridview.hh>
 
 #include "mmiterator.hh"
+#include "multiindexset.hh"
 
 namespace Dune
 {
-  template <class HostGrid>
-  class MultiIndexSet
-  {
-  public:
-    using LeafIndexSet = typename HostGrid::Traits::LeafIndexSet;
-    using LevelIndexSet = typename HostGrid::Traits::LevelIndexSet;
-
-    using IndexType = std::common_type_t<typename LeafIndexSet::IndexType, typename LevelIndexSet::IndexType>;
-
-    using IndexSetTypes = std::variant<LeafIndexSet const*, LevelIndexSet const*>;
-
-    template <class IndexSet>
-    MultiIndexSet(IndexSet const* indexSet)
-      : indexSets_(indexSet)
-    {}
-
-    MultiIndexSet(IndexSetTypes const& indexSet)
-      : indexSets_(indexSet)
-    {}
-
-    template <class Entity>
-    IndexType index (const Entity& e) const
-    {
-      return std::visit([&e](auto const* is) -> IndexType { return is->index(e); }, indexSets_);
-    }
-
-    template <class Entity>
-    IndexType subIndex (const Entity& e, int i, unsigned int codim) const
-    {
-      return std::visit([&e,i,codim](auto const* is) -> IndexType { return is->subIndex(e,i,codim); }, indexSets_);
-    }
-
-  private:
-    IndexSetTypes indexSets_;
-  };
-
-
   // forward declaration
   template <class HostGrid>
   class MultiGridView;
@@ -104,8 +68,9 @@ namespace Dune
     using Grid = typename Traits::Grid;
 
     using GridViewTypes = std::variant<typename HostGrid::LeafGridView, typename HostGrid::LevelGridView>;
+
     using IndexSet = typename Traits::IndexSet;
-    using IndexSetTypes = std::variant<const typename HostGrid::LeafIndexSet*, const typename HostGrid::LevelIndexSet*>;
+    using IndexSetTypes = typename IndexSet::IndexSetTypes;
     using IntersectionIterator = typename Traits::IntersectionIterator;
 
     using CollectiveCommunication = typename Traits::CollectiveCommunication;
@@ -194,14 +159,16 @@ namespace Dune
     // NOTE: IntersectionIterator is one of {LeafIntersectionIterator, LevelIntersectionIterator}
     IntersectionIterator ibegin (std::size_t idx, const typename Codim<0>::Entity& entity) const
     {
-      return std::visit([&entity](auto const& gv) { return gv.ibegin(entity); }, gridViews_[idx]);
+      using II = IntersectionIterator;
+      return std::visit([&entity](auto const& gv) -> II { return gv.ibegin(entity); }, gridViews_[idx]);
     }
 
     /// Obtain end intersection iterator with respect to this view
     // NOTE: IntersectionIterator is one of {LeafIntersectionIterator, LevelIntersectionIterator}
     IntersectionIterator iend (std::size_t idx, const typename Codim<0>::Entity& entity) const
     {
-      return std::visit([&entity](auto const& gv) { return gv.iend(entity); }, gridViews_[idx]);
+      using II = IntersectionIterator;
+      return std::visit([&entity](auto const& gv) -> II { return gv.iend(entity); }, gridViews_[idx]);
     }
 
     /// Obtain collective communication object
diff --git a/dune/multimesh/multiindexset.hh b/dune/multimesh/multiindexset.hh
new file mode 100644
index 0000000000000000000000000000000000000000..1ebfb38a6b88d01dc76558f6d7051d98b85f8c8a
--- /dev/null
+++ b/dune/multimesh/multiindexset.hh
@@ -0,0 +1,113 @@
+// -*- tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 2 -*-
+// vi: set et ts=4 sw=2 sts=2:
+#ifndef DUNE_MULTI_INDEXSET_HH
+#define DUNE_MULTI_INDEXSET_HH
+
+#if ! DUNE_HAVE_CXX_VARIANT
+#error "Require C++17 variant!"
+#endif
+
+#include <type_traits>
+#include <variant>
+
+#include <dune/common/typetraits.hh>
+#include <dune/common/exceptions.hh>
+#include <dune/common/std/type_traits.hh>
+
+#include <dune/grid/common/indexidset.hh>
+
+namespace Dune
+{
+  // forward declaration
+  template <class HostGrid>
+  class MultiGridView;
+
+  template <class HostGrid>
+  class MultiIndexSet
+  {
+  private:
+    using LeafIndexSet = typename HostGrid::Traits::LeafIndexSet;
+    using LevelIndexSet = typename HostGrid::Traits::LevelIndexSet;
+    using IndexSetTypes = std::variant<LeafIndexSet const*, LevelIndexSet const*>;
+
+    friend class MultiGridView<HostGrid>;
+
+  public:
+    using IndexType = typename LeafIndexSet::IndexType;
+    using Types = typename LeafIndexSet::Types;
+
+    static const int dimension = LeafIndexSet::dimension;
+
+  public:
+    template <class IndexSet>
+    MultiIndexSet (IndexSet const* indexSet)
+      : indexSets_(indexSet)
+    {}
+
+    MultiIndexSet (IndexSetTypes const& indexSet)
+      : indexSets_(indexSet)
+    {}
+
+    /// Map entity to index.
+    template <class Entity>
+    IndexType index (const Entity& e) const
+    {
+      return std::visit([&e](auto const* is) { return is->index(e); }, indexSets_);
+    }
+
+    /// Map entity to index.
+    template <int cc>
+    IndexType index (const typename LeafIndexSet::Traits::template Codim<cc>::Entity& e) const
+    {
+      return std::visit([&e](auto const* is) { return is->template index<cc>(e); }, indexSets_);
+    }
+
+    /// Map a subentity to an index.
+    template <class Entity>
+    IndexType subIndex (const Entity& e, int i, unsigned int codim) const
+    {
+      return std::visit([&e,i,codim](auto const* is) { return is->subIndex(e,i,codim); }, indexSets_);
+    }
+
+    /// Map a subentity to an index.
+    template <int cc>
+    IndexType subIndex (const typename LeafIndexSet::Traits::template Codim<cc>::Entity& e,
+                        int i, unsigned int codim) const
+    {
+      return std::visit([&e,i,codim](auto const* is) { return is->template subIndex<cc>(e,i,codim); }, indexSets_);
+    }
+
+
+    /// Obtain all geometry types of entities in domain
+    Types types (int codim) const
+    {
+      return std::visit([codim](auto const* is) { return is->types(codim); }, indexSets_);
+    }
+
+    /// Return total number of entities of given geometry type in entity set \f$E\f$.
+    IndexType size (GeometryType type) const
+    {
+      return std::visit([&type](auto const* is) { return is->size(type); }, indexSets_);
+    }
+
+    /// Return total number of entities of given codim in the entity set \f$E\f$. This
+    /// is simply a sum over all geometry types.
+    IndexType size (int codim) const
+    {
+      return std::visit([codim](auto const* is) { return is->size(codim); }, indexSets_);
+    }
+
+    /// Return true if the given entity is contained in \f$E\f$.
+    template <class Entity>
+    bool contains (const Entity& e) const
+    {
+      return std::visit([&e](auto const* is) { return is->contains(e); }, indexSets_);
+    }
+
+  private:
+    IndexSetTypes indexSets_;
+  };
+
+} // end namespace Dune
+
+#endif // DUNE_MULTI_INDEXSET_HH
diff --git a/dune/multimesh/utility/structuredgridbuilder.hh b/dune/multimesh/utility/structuredgridbuilder.hh
index 4d90433ffea0f723e7b6ba2b107d4292b77c593c..db7d1b619c66e5bc3b4fa59477b06c11ef06ed2a 100644
--- a/dune/multimesh/utility/structuredgridbuilder.hh
+++ b/dune/multimesh/utility/structuredgridbuilder.hh
@@ -37,7 +37,7 @@ namespace Dune
      *  \param upperRight Upper right corner of the grid
      *  \param elements Number of elements in each coordinate direction
      **/
-    static std::unique_ptr<GridType> createCubeGrid (
+    static void createCubeGrid (
         GridFactory<GridType>& factory,
         const FieldVector<ctype,dimworld>& lowerLeft,
         const FieldVector<ctype,dimworld>& upperRight,
@@ -88,9 +88,6 @@ namespace Dune
           factory.insertElement(GeometryTypes::cube(dim), corners);
         }
       }
-
-      // Create the grid and hand it to the calling method
-      return std::unique_ptr<GridType>(factory.createGrid());
     }
 
     static std::unique_ptr<GridType> createCubeGrid (
@@ -99,7 +96,8 @@ namespace Dune
         const std::array<unsigned int,dim>& elements)
     {
       GridFactory<GridType> factory;
-      return createCubeGrid(factory, lowerLeft, upperRight, elements);
+      createCubeGrid(factory, lowerLeft, upperRight, elements);
+      return std::unique_ptr<GridType>(factory.createGrid());
     }
 
     /// \brief Create a structured simplex grid