diff --git a/test/test_global_qbx.py b/test/test_global_qbx.py
index e7a4eaa6baa29df83e22329e4c585b7525b00b80..d0128819b8d01d3c3d6c33b84f7848f9b6a5a9a7 100644
--- a/test/test_global_qbx.py
+++ b/test/test_global_qbx.py
@@ -217,13 +217,7 @@ def test_source_refinement_3d(ctx_getter, surface_name, surface_f, order):
 
 
 @pytest.mark.parametrize(("curve_name", "curve_f", "nelements"), [
-    # TODO: This used to pass for a 20-to-1 ellipse (with different center
-    # placement).  It still produces valid output right now, but the current
-    # test is not smart enough to recognize it. It might be useful to fix that.
-    #
-    # See discussion at
-    # https://gitlab.tiker.net/inducer/pytential/merge_requests/95#note_24003
-    ("18-to-1 ellipse", partial(ellipse, 18), 100),
+    ("20-to-1 ellipse", partial(ellipse, 20), 100),
     ("horseshoe", horseshoe, 64),
     ])
 def test_target_association(ctx_getter, curve_name, curve_f, nelements,
@@ -320,6 +314,8 @@ def test_target_association(ctx_getter, curve_name, curve_f, nelements,
 
     expansion_radii = lpot_source._expansion_radii("ncenters").get(queue)
 
+    surf_targets = np.array(
+            [axis.get(queue) for axis in lpot_source.density_discr.nodes()])
     int_targets = np.array([axis.get(queue) for axis in int_targets.nodes()])
     ext_targets = np.array([axis.get(queue) for axis in ext_targets.nodes()])
 
@@ -352,48 +348,45 @@ def test_target_association(ctx_getter, curve_name, curve_f, nelements,
     if visualize:
         visualize_curve_and_assoc()
 
-    # Checks that the sources match with their own centers.
-    def check_on_surface_targets(nsources, true_side, target_to_center,
-                                 target_to_side_result):
-        assert (target_to_center >= 0).all()
-
-        sources = np.arange(0, nsources)
-
-        # Centers are on alternating sides of the geometry. Dividing by
-        # two yields the number of the source that spawned the center.
-        assert (target_to_center//2 == sources).all()
-
-        assert (target_to_side_result == true_side).all()
-
     # Checks that the targets match with centers on the appropriate side and
     # within the allowable distance.
     def check_close_targets(centers, targets, true_side,
                             target_to_center, target_to_side_result,
                             tgt_slice):
-        assert (target_to_center >= 0).all()
+        targets_have_centers = (target_to_center >= 0).all()
+        assert targets_have_centers
+
         assert (target_to_side_result == true_side).all()
+
+        TOL = 1e-3
         dists = la.norm((targets.T - centers.T[target_to_center]), axis=1)
-        assert (dists <= expansion_radii[target_to_center]).all()
+        assert (dists <= (1 + TOL) * expansion_radii[target_to_center]).all()
 
     # Center side order = -1, 1, -1, 1, ...
     target_to_center_side = 2 * (target_assoc.target_to_center % 2) - 1
 
-    check_on_surface_targets(
-        nsources, -1,
+    # interior surface
+    check_close_targets(
+        centers, surf_targets, -1,
         target_assoc.target_to_center[surf_int_slice],
-        target_to_center_side[surf_int_slice])
+        target_to_center_side[surf_int_slice],
+        surf_int_slice)
 
-    check_on_surface_targets(
-        nsources, +1,
+    # exterior surface
+    check_close_targets(
+        centers, surf_targets, +1,
         target_assoc.target_to_center[surf_ext_slice],
-        target_to_center_side[surf_ext_slice])
+        target_to_center_side[surf_ext_slice],
+        surf_ext_slice)
 
+    # interior volume
     check_close_targets(
         centers, int_targets, -1,
         target_assoc.target_to_center[vol_int_slice],
         target_to_center_side[vol_int_slice],
         vol_int_slice)
 
+    # exterior volume
     check_close_targets(
         centers, ext_targets, +1,
         target_assoc.target_to_center[vol_ext_slice],
diff --git a/test/test_layer_pot.py b/test/test_layer_pot.py
index 3e50aa94a36d8e5710a3bd3b6b4fbaa95ba855fd..66445fee0eda170998b7be5dccd424d9154357ae 100644
--- a/test/test_layer_pot.py
+++ b/test/test_layer_pot.py
@@ -92,7 +92,7 @@ def test_off_surface_eval(ctx_getter, use_fmm, do_plot=False):
     nelements = 30
     target_order = 8
     qbx_order = 3
-    if use_fmm is True:
+    if use_fmm:
         fmm_order = qbx_order
     else:
         fmm_order = False
@@ -139,8 +139,7 @@ def test_off_surface_eval(ctx_getter, use_fmm, do_plot=False):
         pt.colorbar()
         pt.show()
 
-    # FIXME: Why does the FMM only meet this sloppy tolerance?
-    assert linf_err < 1e-2
+    assert linf_err < 1e-3
 
 # }}}