diff --git a/README.md b/README.md
index ec13b5ff..453498ed 100644
--- a/README.md
+++ b/README.md
@@ -257,6 +257,9 @@ with respect to documented and/or tested features.
 - Added: `ElementTriCCR` and `ElementTetCCR`, conforming Crouzeix-Raviart finite elements
 - Fixed: `Mesh.mirrored` returned a wrong mesh when a point other than the origin was used
 - Fixed: `MeshLine` constructor accepted only NumPy arrays and not plain Python lists
+- Fixed: `Mesh.element_finder` (and `CellBasis.probes`, `CellBasis.interpolator`) was not working properly for a small number of elements (<5) or a large number of input points (>1000)
+- Fixed: `MeshTet` and `MeshTri.element_finder` is are now more robust against degenerate elements
+- Fixed: `Mesh.element_finder` (and `CellBasis.probes`, `CellBasis.interpolator`) raises exception if the query point is outside of the domain
 
 ### [3.1.0] - 2021-06-18
 
diff --git a/docs/examples/ex28.py b/docs/examples/ex28.py
index 5c526815..987ce837 100644
--- a/docs/examples/ex28.py
+++ b/docs/examples/ex28.py
@@ -88,15 +88,7 @@ import numpy as np
 import pygmsh
 
 
-if version.parse(pygmsh.__version__) < version.parse('7.0.0'):
-    class NullContextManager():
-        def __enter__(self):
-            return None
-        def __exit__(self, *args):
-            pass
-    geometrycontext = NullContextManager()
-else:
-    geometrycontext = pygmsh.geo.Geometry()
+geometrycontext = pygmsh.geo.Geometry()
 
 halfheight = 1.
 length = 10.
@@ -110,17 +102,12 @@ peclet = 357.
 def make_mesh(halfheight: float,  # mm
               length: float,
               thickness: float) -> MeshTri:
-    with geometrycontext as g:
-        if version.parse(pygmsh.__version__) < version.parse('7.0.0'):
-            geom = pygmsh.built_in.Geometry()
-            geom.add_curve_loop = geom.add_line_loop
-        else:
-            geom = g
+    with geometrycontext as geom:
 
         points = []
         lines = []
 
-        lcar = halfheight / 2**2
+        lcar = halfheight / 2 ** 2
 
         for xy in [(0., halfheight),
                    (0., -halfheight),
@@ -155,10 +142,7 @@ def make_mesh(halfheight: float,  # mm
         geom.add_physical(geom.add_plane_surface(geom.add_curve_loop(
             [*lines[-3:], -lines[1]])), 'solid')
 
-        if version.parse(pygmsh.__version__) < version.parse('7.0.0'):
-            return from_meshio(pygmsh.generate_mesh(geom, dim=2))
-        else:
-            return from_meshio(geom.generate_mesh(dim=2))
+        return from_meshio(geom.generate_mesh(dim=2))
 
 mesh = from_file(Path(__file__).parent / 'meshes' / 'ex28.json')
 element = ElementTriP1()
diff --git a/skfem/mesh/mesh_line_1.py b/skfem/mesh/mesh_line_1.py
index ef85502c..e04afbda 100644
--- a/skfem/mesh/mesh_line_1.py
+++ b/skfem/mesh/mesh_line_1.py
@@ -74,6 +74,7 @@ class MeshLine1(Mesh):
         return np.max(np.abs(self.p[0, self.t[1]] - self.p[0, self.t[0]]))
 
     def element_finder(self, mapping=None):
+
         ix = np.argsort(self.p[0])
         maxt = self.t[np.argmax(self.p[0, self.t], 0),
                       np.arange(self.t.shape[1])]
@@ -81,8 +82,11 @@ class MeshLine1(Mesh):
         def finder(x):
             xin = x.copy()  # bring endpoint inside for np.digitize
             xin[x == self.p[0, ix[-1]]] = self.p[0, ix[-2:]].mean()
-            return np.nonzero(ix[np.digitize(xin, self.p[0, ix])][:, None]
-                              == maxt)[1]
+            elems = np.nonzero(ix[np.digitize(xin, self.p[0, ix])][:, None]
+                               == maxt)[1]
+            if len(elems) < len(x):
+                raise ValueError("Point is outside of the mesh.")
+            return elems
 
         return finder
 
diff --git a/skfem/mesh/mesh_tet_1.py b/skfem/mesh/mesh_tet_1.py
index a0ef8361..f7f75bbd 100644
--- a/skfem/mesh/mesh_tet_1.py
+++ b/skfem/mesh/mesh_tet_1.py
@@ -34,18 +34,34 @@ class MeshTet1(Mesh3D):
         if mapping is None:
             mapping = self._mapping()
 
-        tree = cKDTree(np.mean(self.p[:, self.t], axis=1).T)
+        if not hasattr(self, '_cached_tree'):
+            self._cached_tree = cKDTree(np.mean(self.p[:, self.t], axis=1).T)
+
+        tree = self._cached_tree
+        nelems = self.t.shape[1]
+
+        def finder(x, y, z, _search_all=False):
+
+            if not _search_all:
+                ix = tree.query(np.array([x, y, z]).T,
+                                min(10, nelems))[1].flatten()
+                _, ix_ind = np.unique(ix, return_index=True)
+                ix = ix[np.sort(ix_ind)]
+            else:
+                ix = np.arange(nelems, dtype=np.int64)
 
-        def finder(x, y, z):
-            ix = tree.query(np.array([x, y, z]).T, 5)[1].flatten()
             X = mapping.invF(np.array([x, y, z])[:, None], ix)
-            inside = (
-                (X[0] >= 0) *
-                (X[1] >= 0) *
-                (X[2] >= 0) *
-                (1 - X[0] - X[1] - X[2] >= 0)
-            )
-            return np.array([ix[np.argmax(inside, axis=0)]]).flatten()
+            inside = ((X[0] >= 0) *
+                      (X[1] >= 0) *
+                      (X[2] >= 0) *
+                      (1 - X[0] - X[1] - X[2] >= 0))
+
+            if not inside.max(axis=0).all():
+                if _search_all:
+                    raise ValueError("Point is outside of the mesh.")
+                return finder(x, y, z, _search_all=True)
+
+            return np.array([ix[inside.argmax(axis=0)]]).flatten()
 
         return finder
 
diff --git a/skfem/mesh/mesh_tri_1.py b/skfem/mesh/mesh_tri_1.py
index 055b758a..4b1a087d 100644
--- a/skfem/mesh/mesh_tri_1.py
+++ b/skfem/mesh/mesh_tri_1.py
@@ -325,16 +325,32 @@ class MeshTri1(Mesh2D):
         if mapping is None:
             mapping = self._mapping()
 
-        tree = cKDTree(np.mean(self.p[:, self.t], axis=1).T)
+        if not hasattr(self, '_cached_tree'):
+            self._cached_tree = cKDTree(np.mean(self.p[:, self.t], axis=1).T)
+
+        tree = self._cached_tree
+        nelems = self.t.shape[1]
+
+        def finder(x, y, _search_all=False):
+
+            if not _search_all:
+                ix = tree.query(np.array([x, y]).T,
+                                min(5, nelems))[1].flatten()
+                _, ix_ind = np.unique(ix, return_index=True)
+                ix = ix[np.sort(ix_ind)]
+            else:
+                ix = np.arange(nelems, dtype=np.int64)
 
-        def finder(x, y):
-            ix = tree.query(np.array([x, y]).T, 5)[1].flatten()
             X = mapping.invF(np.array([x, y])[:, None], ix)
-            inside = (
-                (X[0] >= 0) *
-                (X[1] >= 0) *
-                (1 - X[0] - X[1] >= 0)
-            )
-            return np.array([ix[np.argmax(inside, axis=0)]]).flatten()
+            inside = ((X[0] >= 0) *
+                      (X[1] >= 0) *
+                      (1 - X[0] - X[1] >= 0))
+
+            if not inside.max(axis=0).all():
+                if _search_all:
+                    raise ValueError("Point is outside of the mesh.")
+                return finder(x, y, _search_all=True)
+
+            return np.array([ix[inside.argmax(axis=0)]]).flatten()
 
         return finder

diff --git a/tests/test_basis.py b/tests/test_basis.py
index 45676969..9116e431 100644
--- a/tests/test_basis.py
+++ b/tests/test_basis.py
@@ -6,7 +6,7 @@ from numpy.testing import assert_allclose, assert_almost_equal
 
 from skfem import BilinearForm, asm, solve, condense, projection
 from skfem.mesh import MeshTri, MeshTet, MeshHex, MeshQuad, MeshLine
-from skfem.assembly import InteriorBasis, FacetBasis, Dofs, Functional
+from skfem.assembly import CellBasis, FacetBasis, Dofs, Functional
 from skfem.element import (ElementVectorH1, ElementTriP2, ElementTriP1,
                            ElementTetP2, ElementHexS2, ElementHex2,
                            ElementQuad2, ElementLineP2, ElementTriP0,
@@ -28,7 +28,7 @@ class TestCompositeSplitting(TestCase):
 
         e = ElementVectorH1(ElementTriP2()) * ElementTriP1()
 
-        basis = InteriorBasis(m, e)
+        basis = CellBasis(m, e)
 
         @BilinearForm
         def bilinf(u, p, v, q, w):
@@ -93,7 +93,7 @@ class TestFacetExpansion(TestCase):
 
         m = self.mesh_type().refined(2)
 
-        basis = InteriorBasis(m, self.elem_type())
+        basis = CellBasis(m, self.elem_type())
 
         for fun in [lambda x: x[0] == 0,
                     lambda x: x[0] == 1,
@@ -128,7 +128,7 @@ class TestInterpolatorTet(TestCase):
 
     def runTest(self):
         m = self.mesh_type().refined(self.nrefs)
-        basis = InteriorBasis(m, self.element_type())
+        basis = CellBasis(m, self.element_type())
         x = projection(lambda x: x[0] ** 2, basis)
         fun = basis.interpolator(x)
         X = np.linspace(0, 1, 10)
@@ -187,7 +187,38 @@ class TestIncompatibleMeshElement(TestCase):
         with self.assertRaises(ValueError):
             m = MeshTri()
             e = ElementTetP2()
-            basis = InteriorBasis(m, e)
+            basis = CellBasis(m, e)
+
+
+@pytest.mark.parametrize(
+    "mtype,e,nrefs,npoints",
+    [
+        (MeshTri, ElementTriP1(), 0, 10),
+        (MeshTri, ElementTriP2(), 1, 10),
+        (MeshTri, ElementTriP1(), 5, 10),
+        (MeshTri, ElementTriP1(), 1, 3e5),
+        (MeshTet, ElementTetP2(), 1, 10),
+        (MeshTet, ElementTetP1(), 5, 10),
+        (MeshTet, ElementTetP1(), 1, 3e5),
+        (MeshQuad, ElementQuad1(), 1, 10),
+        (MeshQuad, ElementQuad1(), 1, 3e5),
+        (MeshHex, ElementHex1(), 1, 1e5),
+    ]
+)
+def test_interpolator_probes(mtype, e, nrefs, npoints):
+
+    m = mtype().refined(nrefs)
+
+    np.random.seed(0)
+    X = np.random.rand(m.p.shape[0], int(npoints))
+
+    basis = CellBasis(m, e)
+
+    y = projection(lambda x: x[0] ** 2, basis)
+
+    assert_allclose(basis.probes(X) @ y, basis.interpolator(y)(X))
+    atol = 1e-1 if nrefs <= 1 else 1e-3
+    assert_allclose(basis.probes(X) @ y, X[0] ** 2, atol=atol)
 
 
 @pytest.mark.parametrize(
@@ -215,7 +246,7 @@ def test_trace(mtype, e1, e2):
     # use the boundary where last coordinate is zero
     basis = FacetBasis(m, e1,
                        facets=m.facets_satisfying(lambda x: x[x.shape[0] - 1] == 0.0))
-    xfun = projection(lambda x: x[0], InteriorBasis(m, e1))
+    xfun = projection(lambda x: x[0], CellBasis(m, e1))
     nbasis, y = basis.trace(xfun, lambda p: p[0:(p.shape[0] - 1)], target_elem=e2)
 
     @Functional
@@ -235,8 +266,8 @@ def test_point_source(etype):
     from skfem.models.poisson import laplace
 
     mesh = MeshLine().refined()
-    basis = InteriorBasis(mesh, etype())
+    basis = CellBasis(mesh, etype())
     source = np.array([0.7])
     u = solve(*condense(asm(laplace, basis), basis.point_source(source), D=basis.find_dofs()))
     exact = np.stack([(1 - source) * mesh.p, (1 - mesh.p) * source]).min(0)
-    assert_almost_equal(u[basis.nodal_dofs], exact)
\ No newline at end of file
+    assert_almost_equal(u[basis.nodal_dofs], exact)
diff --git a/tests/test_mesh.py b/tests/test_mesh.py
index 611285ef..12380427 100644
--- a/tests/test_mesh.py
+++ b/tests/test_mesh.py
@@ -2,10 +2,12 @@ from unittest import TestCase
 from pathlib import Path
 
 import numpy as np
-from numpy.testing import assert_array_equal
 import pytest
+from scipy.spatial import Delaunay
+from numpy.testing import assert_array_equal
 
-from skfem.mesh import Mesh, MeshHex, MeshLine, MeshQuad, MeshTet, MeshTri, MeshTri2, MeshQuad2, MeshTet2, MeshHex2
+from skfem.mesh import (Mesh, MeshHex, MeshLine, MeshQuad, MeshTet, MeshTri,
+                        MeshTri2, MeshQuad2, MeshTet2, MeshHex2)
 from skfem.io.meshio import to_meshio, from_meshio
 
 
@@ -241,6 +243,34 @@ class TestFinder1DLinspaced(TestCase):
             self.assertEqual(finder(np.array([0.001]))[0], 0)
 
 
+
+@pytest.mark.parametrize(
+    "m,seed",
+    [
+        (MeshTri(), 0),
+        (MeshTri(), 1),
+        (MeshTri(), 2),
+        (MeshTet(), 0),
+        (MeshTet(), 1),
+        (MeshTet(), 2),
+        (MeshTet(), 10),
+    ]
+)
+def test_finder_simplex(m, seed):
+
+    np.random.seed(seed)
+    points = np.hstack((m.p, np.random.rand(m.p.shape[0], 100)))
+    tri = Delaunay(points.T)
+    M = type(m)(points, tri.simplices.T)
+    finder = M.element_finder()
+    
+    query_pts = np.random.rand(m.p.shape[0], 500)
+    assert_array_equal(
+        tri.find_simplex(query_pts.T),
+        finder(*query_pts),
+    )
+
+
 @pytest.mark.parametrize(
     "m",
     [

Examples providing your own mesh and retriving the basis functions?
Are there any examples for supplying your own mesh (for example from scipy Delaunay) and extracting basis functions of some degree? 

For example you might evaluate the basis functions at a set of points for fitting against data as a kind of smoothing spline.

tests/test_basis.py::TestCompositeSplitting::runTest tests/test_basis.py::TestCompositeFacetAssembly::runTest tests/test_basis.py::TestFacetExpansion::runTest tests/test_basis.py::TestFacetExpansionHexS2::runTest tests/test_basis.py::TestFacetExpansionHex2::runTest tests/test_basis.py::TestInterpolatorTet::runTest tests/test_basis.py::TestInterpolatorTet2::runTest tests/test_basis.py::TestInterpolatorTri::runTest tests/test_basis.py::TestInterpolatorQuad::runTest tests/test_basis.py::TestInterpolatorHex::runTest tests/test_basis.py::TestInterpolatorLine::runTest tests/test_basis.py::TestInterpolatorLine2::runTest tests/test_basis.py::TestIncompatibleMeshElement::runTest tests/test_basis.py::test_interpolator_probes[MeshTri1-e0-0-10] tests/test_basis.py::test_interpolator_probes[MeshTri1-e1-1-10] tests/test_basis.py::test_interpolator_probes[MeshTri1-e2-5-10] tests/test_basis.py::test_interpolator_probes[MeshTri1-e3-1-300000.0] tests/test_basis.py::test_interpolator_probes[MeshTet1-e4-1-10] tests/test_basis.py::test_interpolator_probes[MeshTet1-e5-5-10] tests/test_basis.py::test_interpolator_probes[MeshTet1-e6-1-300000.0] tests/test_basis.py::test_interpolator_probes[MeshQuad1-e7-1-10] tests/test_basis.py::test_interpolator_probes[MeshQuad1-e8-1-300000.0] tests/test_basis.py::test_interpolator_probes[MeshHex1-e9-1-100000.0] tests/test_basis.py::test_trace[MeshTri1-e10-e20] tests/test_basis.py::test_trace[MeshTri1-e11-e21] tests/test_basis.py::test_trace[MeshTri1-e12-e22] tests/test_basis.py::test_trace[MeshTri1-e13-e23] tests/test_basis.py::test_trace[MeshTri1-e14-None] tests/test_basis.py::test_trace[MeshQuad1-e15-e25] tests/test_basis.py::test_trace[MeshQuad1-e16-e26] tests/test_basis.py::test_trace[MeshQuad1-e17-e27] tests/test_basis.py::test_trace[MeshTet1-e18-e28] tests/test_basis.py::test_trace[MeshTet1-e19-e29] tests/test_basis.py::test_trace[MeshHex1-e110-e210] tests/test_basis.py::test_trace[MeshHex1-e111-e211] tests/test_basis.py::test_trace[MeshHex1-e112-e212] tests/test_basis.py::test_point_source[ElementLineP1] tests/test_basis.py::test_point_source[ElementLineP2] tests/test_basis.py::test_point_source[ElementLineMini] tests/test_mesh.py::MeshTests::runTest tests/test_mesh.py::Loading::runTest tests/test_mesh.py::SerializeUnserializeCycle::runTest tests/test_mesh.py::TestBoundaryEdges::runTest tests/test_mesh.py::TestBoundaryEdges2::runTest tests/test_mesh.py::TestMeshAddition::runTest tests/test_mesh.py::TestMeshQuadSplit::runTest tests/test_mesh.py::TestAdaptiveSplitting1D::runTest tests/test_mesh.py::TestAdaptiveSplitting2D::runTest tests/test_mesh.py::TestFinder1DRefined::runTest tests/test_mesh.py::TestFinder1DLinspaced::runTest tests/test_mesh.py::test_finder_simplex[m0-0] tests/test_mesh.py::test_finder_simplex[m1-1] tests/test_mesh.py::test_finder_simplex[m2-2] tests/test_mesh.py::test_finder_simplex[m3-0] tests/test_mesh.py::test_finder_simplex[m4-1] tests/test_mesh.py::test_finder_simplex[m5-2] tests/test_mesh.py::test_finder_simplex[m6-10] tests/test_mesh.py::test_meshio_cycle[m0] tests/test_mesh.py::test_meshio_cycle[m1] tests/test_mesh.py::test_meshio_cycle[m2] tests/test_mesh.py::test_meshio_cycle[m3] tests/test_mesh.py::test_meshio_cycle[m4] tests/test_mesh.py::test_meshio_cycle[m5] tests/test_mesh.py::test_meshio_cycle[m6] tests/test_mesh.py::test_saveload_cycle[m0] tests/test_mesh.py::test_saveload_cycle[m1] tests/test_mesh.py::test_saveload_cycle[m2] tests/test_mesh.py::test_saveload_cycle[m3]