diff --git a/test/merged-cubes.step b/test/merged-cubes.step
new file mode 100644
index 0000000000000000000000000000000000000000..7e9b273b3b313c35d70c2c216fd1ef5fb6c44169
--- /dev/null
+++ b/test/merged-cubes.step
@@ -0,0 +1,4065 @@
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+#3188 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
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+#3200 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
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+#3214 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
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+#3221 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
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+#3245 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
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+#3272 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
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+#3279 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
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+#3293 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
+#3294 = PCURVE('',#3190,#3295);
+#3295 = DEFINITIONAL_REPRESENTATION('',(#3296),#3300);
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+#3297 = CARTESIAN_POINT('',(-0.,0.));
+#3298 = VECTOR('',#3299,1.);
+#3299 = DIRECTION('',(-0.,1.));
+#3300 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
+#3301 = ORIENTED_EDGE('',*,*,#3044,.T.);
+#3302 = ADVANCED_FACE('',(#3303),#2982,.T.);
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+#3318 = CARTESIAN_POINT('',(1.570796326795,0.));
+#3319 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
+#3320 = PCURVE('',#3190,#3321);
+#3321 = DEFINITIONAL_REPRESENTATION('',(#3322),#3325);
+#3322 = B_SPLINE_CURVE_WITH_KNOTS('',1,(#3323,#3324),.UNSPECIFIED.,.F.,
+  .F.,(2,2),(4.712388980385,6.28318530718),.PIECEWISE_BEZIER_KNOTS.);
+#3323 = CARTESIAN_POINT('',(-1.570796326795,8.));
+#3324 = CARTESIAN_POINT('',(0.,8.));
+#3325 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
+#3326 = ORIENTED_EDGE('',*,*,#3020,.F.);
+#3327 = ADVANCED_FACE('',(#3328),#3103,.T.);
+#3328 = FACE_BOUND('',#3329,.T.);
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+#3342 = CARTESIAN_POINT('',(0.,2.));
+#3343 = VECTOR('',#3344,1.);
+#3344 = DIRECTION('',(1.,0.));
+#3345 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
+#3346 = PCURVE('',#3347,#3352);
+#3347 = SPHERICAL_SURFACE('',#3348,2.);
+#3348 = AXIS2_PLACEMENT_3D('',#3349,#3350,#3351);
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+#3355 = VECTOR('',#3356,1.);
+#3356 = DIRECTION('',(1.,0.));
+#3357 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
+#3358 = ORIENTED_EDGE('',*,*,#3202,.T.);
+#3359 = ORIENTED_EDGE('',*,*,#3089,.F.);
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+#3367 = AXIS2_PLACEMENT_3D('',#3368,#3369,#3370);
+#3368 = CARTESIAN_POINT('',(-5.5,-5.5,-5.5));
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+#3376 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
+#3377 = PCURVE('',#3347,#3378);
+#3378 = DEFINITIONAL_REPRESENTATION('',(#3379),#3382);
+#3379 = B_SPLINE_CURVE_WITH_KNOTS('',1,(#3380,#3381),.UNSPECIFIED.,.F.,
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+#3381 = CARTESIAN_POINT('',(0.,0.));
+#3382 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
+#3383 = ORIENTED_EDGE('',*,*,#3258,.T.);
+#3384 = ORIENTED_EDGE('',*,*,#3227,.T.);
+#3385 = ORIENTED_EDGE('',*,*,#3143,.F.);
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+#3387 = FACE_BOUND('',#3388,.F.);
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+#3399 = B_SPLINE_CURVE_WITH_KNOTS('',1,(#3400,#3401),.UNSPECIFIED.,.F.,
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+#3401 = CARTESIAN_POINT('',(-1.570796326795,2.));
+#3402 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
+#3403 = PCURVE('',#3347,#3404);
+#3404 = DEFINITIONAL_REPRESENTATION('',(#3405),#3408);
+#3405 = B_SPLINE_CURVE_WITH_KNOTS('',1,(#3406,#3407),.UNSPECIFIED.,.F.,
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+#3407 = CARTESIAN_POINT('',(1.570796326795,0.));
+#3408 = ( GEOMETRIC_REPRESENTATION_CONTEXT(2) 
+PARAMETRIC_REPRESENTATION_CONTEXT() REPRESENTATION_CONTEXT('2D SPACE',''
+  ) );
+#3409 = ORIENTED_EDGE('',*,*,#3174,.T.);
+#3410 = ORIENTED_EDGE('',*,*,#3307,.T.);
+#3411 = ORIENTED_EDGE('',*,*,#3281,.F.);
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+#3416 = ORIENTED_EDGE('',*,*,#3390,.T.);
+#3417 = ORIENTED_EDGE('',*,*,#3332,.F.);
+#3418 = ( GEOMETRIC_REPRESENTATION_CONTEXT(3) 
+GLOBAL_UNCERTAINTY_ASSIGNED_CONTEXT((#3422)) 
+GLOBAL_UNIT_ASSIGNED_CONTEXT((#3419,#3420,#3421)) REPRESENTATION_CONTEXT
+('Context #1','3D Context with UNIT and UNCERTAINTY') );
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+#3422 = UNCERTAINTY_MEASURE_WITH_UNIT(LENGTH_MEASURE(1.E-07),#3419,
+  'distance_accuracy_value','confusion accuracy');
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+#3428 = MECHANICAL_CONTEXT('',#2,'mechanical');
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+#3431 = ( REPRESENTATION_RELATIONSHIP('','',#25,#10) 
+REPRESENTATION_RELATIONSHIP_WITH_TRANSFORMATION(#3432) 
+SHAPE_REPRESENTATION_RELATIONSHIP() );
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+#3433 = PRODUCT_DEFINITION_SHAPE('Placement','Placement of an item',
+  #3434);
+#3434 = NEXT_ASSEMBLY_USAGE_OCCURRENCE('5','=>[0:1:1:2]','',#5,#3425,$);
+#3435 = PRODUCT_TYPE('part',$,(#3427));
+#3436 = MECHANICAL_DESIGN_GEOMETRIC_PRESENTATION_REPRESENTATION('',(
+    #3437),#3418);
+#3437 = STYLED_ITEM('color',(#3438),#26);
+#3438 = PRESENTATION_STYLE_ASSIGNMENT((#3439,#3445));
+#3439 = SURFACE_STYLE_USAGE(.BOTH.,#3440);
+#3440 = SURFACE_SIDE_STYLE('',(#3441));
+#3441 = SURFACE_STYLE_FILL_AREA(#3442);
+#3442 = FILL_AREA_STYLE('',(#3443));
+#3443 = FILL_AREA_STYLE_COLOUR('',#3444);
+#3444 = COLOUR_RGB('',0.800000011921,0.800000011921,0.800000011921);
+#3445 = CURVE_STYLE('',#3446,POSITIVE_LENGTH_MEASURE(0.1),#3444);
+#3446 = DRAUGHTING_PRE_DEFINED_CURVE_FONT('continuous');
+ENDSEC;
+END-ISO-10303-21;
diff --git a/test/test_layer_pot.py b/test/test_layer_pot.py
index 7e86a96ebcbc28d84cfaf218f8957c3b1296d1d0..0f3d35bb9394359433c376394303121884456a63 100644
--- a/test/test_layer_pot.py
+++ b/test/test_layer_pot.py
@@ -304,20 +304,22 @@ def run_int_eq_test(cl_ctx, queue, case, resolution):
 
     refiner_extra_kwargs = {}
 
-    if case.k != 0:
-        refiner_extra_kwargs["kernel_length_scale"] = 5/case.k
-
     if case.fmm_backend is None:
         fmm_order = False
     else:
-        fmm_order = case.qbx_order + 5
+        fmm_order = case.fmm_order
 
     qbx = QBXLayerPotentialSource(
             pre_density_discr, fine_order=source_order, qbx_order=case.qbx_order,
             fmm_order=fmm_order, fmm_backend=case.fmm_backend)
 
     if case.use_refinement:
+        if case.k != 0:
+            refiner_extra_kwargs["kernel_length_scale"] = 5/case.k
+
+        print("%d elements before refinement" % pre_density_discr.mesh.nelements)
         qbx, _ = qbx.with_refinement(**refiner_extra_kwargs)
+        print("%d elements after refinement" % qbx.density_discr.mesh.nelements)
 
     density_discr = qbx.density_discr
 
@@ -336,12 +338,12 @@ def run_int_eq_test(cl_ctx, queue, case, resolution):
 
         elif mesh.ambient_dim == 3:
             from meshmode.discretization.visualization import make_visualizer
-            bdry_vis = make_visualizer(queue, density_discr, case.target_order)
+            bdry_vis = make_visualizer(queue, density_discr, case.target_order + 5)
 
             bdry_normals = bind(density_discr, sym.normal(3))(queue)\
                     .as_vector(dtype=object)
 
-            bdry_vis.write_vtk_file("source-%s.vtu" % resolution, [
+            bdry_vis.write_vtk_file("pre-solve-source-%s.vtu" % resolution, [
                 ("bdry_normals", bdry_normals),
                 ])
 
@@ -544,7 +546,7 @@ def run_int_eq_test(cl_ctx, queue, case, resolution):
 
     if 0:
         from meshmode.discretization.visualization import make_visualizer
-        bdry_vis = make_visualizer(queue, density_discr, case.target_order)
+        bdry_vis = make_visualizer(queue, density_discr, case.target_order+5)
 
         bdry_normals = bind(density_discr, sym.normal(3))(queue)\
                 .as_vector(dtype=object)
@@ -736,7 +738,6 @@ class CurveIntEqTestCase(IntEqTestCase):
                 np.linspace(0, 1, resolution+1),
                 target_order)
 
-    fmm_backend = None
     use_refinement = True
     neumann_alpha = None  # default
 
@@ -745,6 +746,7 @@ class CurveIntEqTestCase(IntEqTestCase):
 
     qbx_order = 5
     target_order = qbx_order
+    fmm_backend = None
 
     check_tangential_deriv = True
 
@@ -756,7 +758,22 @@ class EllipseIntEqTestCase(CurveIntEqTestCase):
         return ellipse(3, x)
 
 
-class EllipsoidIntEqTestCase(IntEqTestCase):
+class Helmholtz3DIntEqTestCase(IntEqTestCase):
+    fmm_backend = "fmmlib"
+    use_refinement = False
+    neumann_alpha = 0  # no double layers in FMMlib backend yet
+
+    qbx_order = 2
+    fmm_order = 7
+    target_order = qbx_order
+    check_tangential_deriv = False
+
+    # We're only expecting three digits based on FMM settings. Who are we
+    # kidding?
+    gmres_tol = 1e-5
+
+
+class EllipsoidIntEqTestCase(Helmholtz3DIntEqTestCase):
     resolutions = [2, 1]
     name = "ellipsoid"
 
@@ -772,20 +789,78 @@ class EllipsoidIntEqTestCase(IntEqTestCase):
         # Flip elements--gmsh generates inside-out geometry.
         return perform_flips(mesh, np.ones(mesh.nelements))
 
-    fmm_backend = "fmmlib"
-    use_refinement = False
-    neumann_alpha = 0  # no double layers in FMMlib backend yet
+    fmm_order = 13
 
     inner_radius = 0.4
     outer_radius = 5
 
-    qbx_order = 2
-    target_order = qbx_order
-    check_tangential_deriv = False
 
-    # We're only expecting three digits based on FMM settings. Who are we
-    # kidding?
-    gmres_tol = 1e-5
+class MergedCubesIntEqTestCase(Helmholtz3DIntEqTestCase):
+    resolutions = [1.4]
+    name = "merged-cubes"
+
+    def get_mesh(self, resolution, target_order):
+        from meshmode.mesh.io import generate_gmsh, FileSource
+        mesh = generate_gmsh(
+                FileSource("merged-cubes.step"), 2, order=2,
+                other_options=[
+                    "-string",
+                    "Mesh.CharacteristicLengthMax = %g;" % resolution])
+
+        from meshmode.mesh.processing import perform_flips
+        # Flip elements--gmsh generates inside-out geometry.
+        mesh = perform_flips(mesh, np.ones(mesh.nelements))
+
+        return mesh
+
+    use_refinement = True
+
+    inner_radius = 0.4
+    outer_radius = 12
+
+
+class ManyEllipsoidIntEqTestCase(Helmholtz3DIntEqTestCase):
+    resolutions = [2, 1]
+    name = "ellipsoid"
+
+    nx = 2
+    ny = 2
+    nz = 2
+
+    def get_mesh(self, resolution, target_order):
+        from meshmode.mesh.io import generate_gmsh, FileSource
+        base_mesh = generate_gmsh(
+                FileSource("ellipsoid.step"), 2, order=2,
+                other_options=[
+                    "-string",
+                    "Mesh.CharacteristicLengthMax = %g;" % resolution])
+
+        from meshmode.mesh.processing import perform_flips
+        # Flip elements--gmsh generates inside-out geometry.
+        base_mesh = perform_flips(base_mesh, np.ones(base_mesh.nelements))
+
+        from meshmode.mesh.processing import affine_map, merge_disjoint_meshes
+        from meshmode.mesh.tools import rand_rotation_matrix
+        pitch = 10
+        meshes = [
+                affine_map(
+                    base_mesh,
+                    A=rand_rotation_matrix(3),
+                    b=pitch*np.array([
+                        (ix-self.nx//2),
+                        (iy-self.ny//2),
+                        (iz-self.ny//2)]))
+                for ix in range(self.nx)
+                for iy in range(self.ny)
+                for iz in range(self.nz)
+                ]
+
+        mesh = merge_disjoint_meshes(meshes, single_group=True)
+        return mesh
+
+    inner_radius = 0.4
+    # This should sit in the area just outside the middle ellipsoid
+    outer_radius = 5
 
 
 @pytest.mark.parametrize("case", [