diff --git a/applications/test/globalIndex2/Test-globalIndex2.cxx b/applications/test/globalIndex2/Test-globalIndex2.cxx
index a8bd73c394..a2df1c3882 100644
--- a/applications/test/globalIndex2/Test-globalIndex2.cxx
+++ b/applications/test/globalIndex2/Test-globalIndex2.cxx
@@ -42,6 +42,21 @@ Description
 using namespace Foam;
 
 
+void printTest1
+(
+    const globalIndex& gi,
+    const label proci,
+    const label value
+)
+{
+    // With range check
+    Info<< "  value:" << value << " on:" << gi.findProc(proci, value)
+        << " below/above: ("
+        << gi.findProcBelow(proci, value) << ' '
+        << gi.findProcAbove(proci, value) << ')' << nl;
+}
+
+
 // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 //  Main program:
 
@@ -87,6 +102,52 @@ int main(int argc, char *argv[])
             << nl;
     }
 
+    {
+        // From some offsets
+        globalIndex gi;
+
+        globalIndex gi0
+        (
+            labelList({ 0, 10, 20, 30, 40, 50, 60 })
+        );
+
+        Info<< "offsets: " << gi0.offsets() << nl;
+
+        // Alternative to copy assigment
+        gi.reset(gi0);
+
+        Info<< "globalIndex :"; gi.offsets().writeList(Info) << nl;
+
+        // Resizing is fine, but also check the binary search results!
+        gi.resize(10);
+        Info<< "globalIndex :"; gi.offsets().writeList(Info) << nl;
+
+        // NB: these routines are mostly failsafe on bad addresses
+        // for (const label proci : { 4, 8, -1 })
+
+        for (const label proci : { 4 })
+        {
+            Info<< "proc:" << proci
+                << " : [" << gi.localStart(proci)
+                << "," << gi.localEnd(proci) << ")"  << nl;
+
+            for (label i = 0; i < 25; ++i)
+            {
+                const label value = rnd.position<label>(-8, 100);
+
+                printTest1(gi, proci, value);
+            }
+
+            Info<< "other on proc:" << proci << nl;
+            printTest1(gi, proci, gi.localStart(proci));
+            printTest1(gi, proci, gi.localEnd(proci));
+
+            Info<< "other on proc:0" << nl;
+            printTest1(gi, 0, gi.localStart(proci));
+            printTest1(gi, 0, gi.localEnd(proci));
+        }
+    }
+
     Info<< "\nEnd\n" << endl;
     return 0;
 }
diff --git a/src/OpenFOAM/parallel/globalIndex/globalIndex.H b/src/OpenFOAM/parallel/globalIndex/globalIndex.H
index 4656e94396..faad801f7d 100644
--- a/src/OpenFOAM/parallel/globalIndex/globalIndex.H
+++ b/src/OpenFOAM/parallel/globalIndex/globalIndex.H
@@ -321,6 +321,9 @@ public:
 
     // Queries and renumbering
 
+        //- True if contained within the offsets range
+        inline bool contains(const label i) const noexcept;
+
         //- Start of proci data
         inline label localStart(const label proci) const;
 
@@ -360,14 +363,26 @@ public:
         //- From global to local on proci
         inline label toLocal(const label proci, const label i) const;
 
-        //- Which processor does global id come from?
-        //  Does an initial check for isLocal first (assumed to occur
-        //  reasonably frequently) followed by a binary search.
-        //- Fatal for out of range ids (eg, negative or >= totalSize()
-        inline label whichProcID(const label i) const;
+        //- Find processor with specified global id.
+        //- Check proci first, followed by binary search.
+        //  \return the processor number or -1 (not found)
+        inline label findProc(const label proci, const label i) const;
+
+        //- Find processor above proci with specified global id - binary search.
+        //  \return the processor above proci or -1 otherwise
+        //  (including for out-of-range indices)
+        inline label findProcAbove(const label proci, const label i) const;
+
+        //- Find processor below proci with specified global id - binary search.
+        //  Binary search.
+        //  \return the processor below proci or -1 otherwise
+        //  (including for out-of-range indices)
+        inline label findProcBelow(const label proci, const label i) const;
 
         //- Which processor does global id come from?
-        //- Checks proci first.
+        //- Checks proci first (assumed to occur reasonably frequently)
+        //- followed by a binary search.
+        //- Fatal for out-of-range indices
         inline label whichProcID(const label proci, const label i) const;
 
 
@@ -404,10 +419,17 @@ public:
         //  FatalError if not on local processor.
         inline label toLocal(const label i) const;
 
+        //- Which processor does global id come from?
+        //  Uses myProcNo for the initial local check.
+        inline label whichProcID(const label i) const;
+
 
     // Iteration
 
-        //- Forward input iterator with const access
+        //- Forward input iterator with const access that is used to
+        //- iterate across the globalIndex offsets() table.
+        //  The de-referenced value is the range() with (start, size),
+        //  but it also provides separate index, start, size information.
         class const_iterator
         {
             //- The parent class for which this is an iterator
diff --git a/src/OpenFOAM/parallel/globalIndex/globalIndexI.H b/src/OpenFOAM/parallel/globalIndex/globalIndexI.H
index a4cd6f4e12..3658725b99 100644
--- a/src/OpenFOAM/parallel/globalIndex/globalIndexI.H
+++ b/src/OpenFOAM/parallel/globalIndex/globalIndexI.H
@@ -163,6 +163,12 @@ inline Foam::label Foam::globalIndex::end_value() const noexcept
 }
 
 
+inline bool Foam::globalIndex::contains(const label i) const noexcept
+{
+    return (begin_value() <= i) && (i < end_value());
+}
+
+
 inline Foam::label Foam::globalIndex::totalSize() const noexcept
 {
     return end_value();
@@ -295,7 +301,7 @@ inline Foam::labelRange Foam::globalIndex::range() const
 inline bool Foam::globalIndex::isLocal(const label proci, const label i) const
 {
     // range contains()
-    return i >= offsets_[proci] && i < offsets_[proci+1];
+    return (offsets_[proci] <= i) && (i < offsets_[proci+1]);
 }
 
 
@@ -377,16 +383,16 @@ inline void Foam::globalIndex::inplaceToGlobal(labelUList& labels) const
 inline Foam::label
 Foam::globalIndex::toLocal(const label proci, const label i) const
 {
-    const label locali = i - offsets_[proci];
-
-    if (locali < 0 || i >= offsets_[proci+1])
+    // range !contains()
+    if (i < offsets_[proci] || i >= offsets_[proci+1])
     {
         FatalErrorInFunction
-            << "Global " << i << " does not belong on processor "
-            << proci << nl << "Offsets:" << offsets_
+            << "Global id:" << i << " does not belong on processor "
+            << proci << nl
+            << " Offsets:" << offsets_
             << abort(FatalError);
     }
-    return locali;
+    return (i - offsets_[proci]);
 }
 
 
@@ -396,21 +402,109 @@ inline Foam::label Foam::globalIndex::toLocal(const label i) const
 }
 
 
+inline Foam::label Foam::globalIndex::findProc
+(
+    const label proci,
+    const label i
+) const
+{
+    // Simple checks first
+    if
+    (
+        (proci < 0)                     // Invalid proc
+     || (proci+1 > offsets_.size())     // Invalid proc
+     || (i < offsets_.front())          // Below the start
+     || (i >= offsets_.back())          // Beyond the end
+    )
+    {
+        return -1;
+    }
+
+    // Assume that in many cases we have many queries for the local proci,
+    // so query that first but also make it the split point for
+    // restricting the binary searches
+
+    if (isLocal(proci, i))
+    {
+        return proci;
+    }
+
+    if (i < offsets_[proci])
+    {
+        // Can restrict search to procs below proci
+        const labelList::subList slice(offsets_, proci);
+        return findLower(slice, i+1);
+    }
+
+    // Search starts at proci+1 (and above)
+    return findLower(offsets_, i+1, proci+1);
+}
+
+
+inline Foam::label Foam::globalIndex::findProcAbove
+(
+    const label proci,
+    const label i
+) const
+{
+    // Simple checks first
+    if
+    (
+        (proci < 0)                     // Invalid proci
+     || (proci+1 >= offsets_.size())    // Nothing above last proc
+     || (i < offsets_[proci+1])         // Exclude proc-local and below
+     || (i >= offsets_.back())          // Beyond the end
+    )
+    {
+        return -1;
+    }
+
+    // Search starts at proci+1 (and above)
+    return findLower(offsets_, i+1, (proci+1));
+}
+
+
+inline Foam::label Foam::globalIndex::findProcBelow
+(
+    const label proci,
+    const label i
+) const
+{
+    if
+    (
+        (proci <= 0)                    // Nothing below first proc
+     || (proci >= offsets_.size())      // Invalid proci
+     || (i >= offsets_[proci])          // Exclude proc-local and above
+     || (i < offsets_[0])               // Beyond the begin
+    )
+    {
+        return -1;
+    }
+
+    // Restrict search to procs below proci
+    const labelList::subList slice(offsets_, proci);
+    return findLower(slice, i+1);
+}
+
+
+// Note: could split this into failsafe/non-failsafe versions
 inline Foam::label Foam::globalIndex::whichProcID
 (
     const label proci,
     const label i
 ) const
 {
-    if (i < 0 || i >= totalSize())
+    label foundProc = findProc(proci, i);
+
+    if (foundProc < 0)
     {
         FatalErrorInFunction
-            << "Global " << i << " does not belong on any processor."
-            << " Offsets:" << offsets_
+            << "Global id:" << i << " does not belong on any processor." << nl
+            << "Offsets:" << offsets_
             << abort(FatalError);
     }
 
-    return isLocal(proci, i) ? proci : findLower(offsets_, i+1);
+    return foundProc;
 }