diff --git a/CHANGELOG.md b/CHANGELOG.md
index edd8768b416477a1a2eb68ba0dbc8969ef843dc5..9e1f0caed09ad206ff4d5e21420098f50934ff7a 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -29,6 +29,8 @@ The new and last argument is the `ResidualType` (see above).
 
 ### Deprecated properties/classes/functions/files, to be removed after 3.7:
 
+- __AMGBackend__: `AMGBiCGSTABBackend` have been deprecated, use `AMGBiCGSTABIstlSolver` instead 
+- __IstlSolverFactoryBackend__: `IstlSolverFactoryBackend` now require an additional parameter `LinearAlgebraTraits`
 
 ### New experimental features (possibly subject to backwards-incompatible changes in the future)
 
diff --git a/test/freeflow/shallowwater/bowl/main.cc b/test/freeflow/shallowwater/bowl/main.cc
index dfd9388c2feeeeb7aa3f1efe6546744960675172..e1b1c448cd9ebdb9a8585fcdc38c74d6008a151d 100644
--- a/test/freeflow/shallowwater/bowl/main.cc
+++ b/test/freeflow/shallowwater/bowl/main.cc
@@ -38,7 +38,8 @@
 
 #include <dumux/io/grid/gridmanager.hh>
 #include <dumux/linear/linearsolvertraits.hh>
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
 #include <dumux/assembly/fvassembler.hh>
@@ -200,7 +201,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/freeflow/shallowwater/poiseuilleflow/main.cc b/test/freeflow/shallowwater/poiseuilleflow/main.cc
index 33c3d9154e400a645f8e87984f4eab76ae0eef9d..3563cbbc9604ff868bf527c55896c5e2f678b00b 100644
--- a/test/freeflow/shallowwater/poiseuilleflow/main.cc
+++ b/test/freeflow/shallowwater/poiseuilleflow/main.cc
@@ -28,7 +28,8 @@
 #include <dumux/common/dumuxmessage.hh>
 
 #include <dumux/linear/linearsolvertraits.hh>
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 #include <dumux/assembly/fvassembler.hh>
 
@@ -106,7 +107,8 @@ int main(int argc, char** argv)
     using Assembler = FVAssembler<TypeTag, DiffMethod::numeric>;
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     using NewtonSolver = Dumux::NewtonSolver<Assembler, LinearSolver>;
diff --git a/test/freeflow/shallowwater/poiseuilleflow/vertical/main.cc b/test/freeflow/shallowwater/poiseuilleflow/vertical/main.cc
index ecde1f9448bcd955c481976cfb9049e7157da6a0..f0eb49655bac47c45c5ff382e964760b629b16e9 100644
--- a/test/freeflow/shallowwater/poiseuilleflow/vertical/main.cc
+++ b/test/freeflow/shallowwater/poiseuilleflow/vertical/main.cc
@@ -28,7 +28,8 @@
 #include <dumux/common/dumuxmessage.hh>
 
 #include <dumux/linear/linearsolvertraits.hh>
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 #include <dumux/assembly/fvassembler.hh>
 
@@ -89,7 +90,8 @@ int main(int argc, char** argv)
     using Assembler = FVAssembler<TypeTag, DiffMethod::numeric>;
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables);
 
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     using NewtonSolver = Dumux::NewtonSolver<Assembler, LinearSolver>;
diff --git a/test/freeflow/shallowwater/roughchannel/main.cc b/test/freeflow/shallowwater/roughchannel/main.cc
index 8fcb0136da3aee0d1bb4b41d4f1e40c1af2e81e9..ef8d37c36b672a5692a3cb62f7d31edb2206914c 100644
--- a/test/freeflow/shallowwater/roughchannel/main.cc
+++ b/test/freeflow/shallowwater/roughchannel/main.cc
@@ -37,7 +37,8 @@
 #include <dumux/common/dumuxmessage.hh>
 
 #include <dumux/io/grid/gridmanager.hh>
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -121,7 +122,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/geomechanics/elastic/main.cc b/test/geomechanics/elastic/main.cc
index d97f25db3c4dc98b23fffdf54d883345d4e01e5e..b8bfc9c5cd4bf353eb8d2df2618d2780c0955591 100644
--- a/test/geomechanics/elastic/main.cc
+++ b/test/geomechanics/elastic/main.cc
@@ -37,7 +37,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -117,7 +118,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/geomechanics/poroelastic/main.cc b/test/geomechanics/poroelastic/main.cc
index 26575be96dc9b346faa3b6551ac80708ac3fd07d..4b1a995c95129a0fc694cf0e62926ee34408352d 100644
--- a/test/geomechanics/poroelastic/main.cc
+++ b/test/geomechanics/poroelastic/main.cc
@@ -38,7 +38,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -185,7 +186,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/linear/test_linearsolver.cc b/test/linear/test_linearsolver.cc
index 441d709893c1b2a72b2e029268eadcc7b941d263..4128648940c8b857952ae384d750241a2321a966 100644
--- a/test/linear/test_linearsolver.cc
+++ b/test/linear/test_linearsolver.cc
@@ -23,7 +23,8 @@
 #include <dumux/linear/linearsolvertraits.hh>
 
 #include <dumux/linear/istlsolverfactorybackend.hh>
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 
 namespace Dumux::Test {
 
@@ -41,7 +42,8 @@ void solveWithFactory(M& A, X& x, V& b, const std::string& paramGroup)
 {
     std::cout << std::endl;
 
-    using LinearSolver = IstlSolverFactoryBackend<LinearSolverTraits<Test::MockGridGeometry>>;
+    using LinearSolver = IstlSolverFactoryBackend<LinearSolverTraits<Test::MockGridGeometry>,
+                                                  LinearAlgebraTraits<M,V>>;
     LinearSolver solver(paramGroup);
 
     std::cout << "Solving Laplace problem with " << solver.name() << "\n";
@@ -76,7 +78,8 @@ int main(int argc, char* argv[])
         std::cout << std::endl;
 
         const auto testSolverName = "AMGBiCGSTAB";
-        using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<Test::MockGridGeometry>>;
+        using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<Test::MockGridGeometry>,
+                                                   LinearAlgebraTraits<Matrix,Vector>>;
         LinearSolver solver(testSolverName);
 
         std::cout << "Solving Laplace problem with " << solver.name() << "\n";
diff --git a/test/porousmediumflow/1p/fracture2d3d/main.cc b/test/porousmediumflow/1p/fracture2d3d/main.cc
index dc518975f3df3c08d5176fffba251814e2046ae9..3fa7a9cd8eeac208b3e91f064247dc919081a4ee 100644
--- a/test/porousmediumflow/1p/fracture2d3d/main.cc
+++ b/test/porousmediumflow/1p/fracture2d3d/main.cc
@@ -39,7 +39,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -123,7 +124,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/1p/isothermal/main.cc b/test/porousmediumflow/1p/isothermal/main.cc
index aa09f83ddaea11ab54698092ad9ead94e7564681..f39b7cf850bd7eec9b2dcbf6e1046a9eaf40d796 100644
--- a/test/porousmediumflow/1p/isothermal/main.cc
+++ b/test/porousmediumflow/1p/isothermal/main.cc
@@ -39,7 +39,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -127,7 +128,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/1p/network1d3d/main.cc b/test/porousmediumflow/1p/network1d3d/main.cc
index 1dae7ad34232f929eb7125ceece0aff263e7e175..4c63af744c098441e7b3a1ace891467c9e39f862 100644
--- a/test/porousmediumflow/1p/network1d3d/main.cc
+++ b/test/porousmediumflow/1p/network1d3d/main.cc
@@ -39,7 +39,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -123,7 +124,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/1p/periodicbc/main.cc b/test/porousmediumflow/1p/periodicbc/main.cc
index 1eb9aaf06d9365c8553930d2314d91b93ad44d24..d4fc147472c4a8e8cfce2f084388d32f25f47047 100644
--- a/test/porousmediumflow/1p/periodicbc/main.cc
+++ b/test/porousmediumflow/1p/periodicbc/main.cc
@@ -29,7 +29,8 @@
 #include <dune/common/parallel/mpihelper.hh>
 #include <dune/common/timer.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 
 #include <dumux/common/initialize.hh>
@@ -125,7 +126,8 @@ int main(int argc, char** argv)
 
     // solve the linear system
     Dune::Timer solverTimer;
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(gridGeometry->gridView(), gridGeometry->dofMapper());
 
     if (mpiHelper.rank() == 0) std::cout << "Solving linear system using " + linearSolver->name() + "..." << std::flush;
diff --git a/test/porousmediumflow/1p/pointsources/timedependent/main.cc b/test/porousmediumflow/1p/pointsources/timedependent/main.cc
index 7c73f8d130b5cceeac9425b92ccb7a62a8956d78..5582fa4b29d5586103398095e1b9ef8f3c12eace 100644
--- a/test/porousmediumflow/1p/pointsources/timedependent/main.cc
+++ b/test/porousmediumflow/1p/pointsources/timedependent/main.cc
@@ -36,7 +36,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -120,7 +121,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/1p/pointsources/timeindependent/main.cc b/test/porousmediumflow/1p/pointsources/timeindependent/main.cc
index 979dc26ae1f0870e1005613f9a3d11ebcda13d43..da33538c8bd21610eb3cb56d9180b82c885c4c2d 100644
--- a/test/porousmediumflow/1p/pointsources/timeindependent/main.cc
+++ b/test/porousmediumflow/1p/pointsources/timeindependent/main.cc
@@ -36,7 +36,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -120,7 +121,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/2p/adaptive/main.cc b/test/porousmediumflow/2p/adaptive/main.cc
index e1949553fe5bd672538bf4db16ace9bcd7b95fb3..582ac592b257fa1a780e87e92c87545b2d50a67a 100644
--- a/test/porousmediumflow/2p/adaptive/main.cc
+++ b/test/porousmediumflow/2p/adaptive/main.cc
@@ -34,7 +34,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -175,7 +176,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
@@ -202,7 +204,7 @@ int main(int argc, char** argv)
                 xOld = x; //!< Overwrite the old solution with the new (resized & interpolated) one
                 gridVariables->updateAfterGridAdaption(x); //!< Initialize the secondary variables to the new (and "new old") solution
                 assembler->updateAfterGridAdaption(); //!< Tell the assembler that the grid changed
-                linearSolver->updateAfterGridAdaption(leafGridView, gridGeometry->dofMapper()); //!< Tell the solver that the grid changed
+                linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper()); //!< Tell the solver that the grid changed
                 problem->computePointSourceMap(); //!< Update the point source map
             }
         }
diff --git a/test/porousmediumflow/2p/cornerpoint/main.cc b/test/porousmediumflow/2p/cornerpoint/main.cc
index e6d0cb061f7f21a02c818b05e4375d17c0185c88..457f52c3c5df5f5e4592b1373598339b97a4abc2 100644
--- a/test/porousmediumflow/2p/cornerpoint/main.cc
+++ b/test/porousmediumflow/2p/cornerpoint/main.cc
@@ -40,7 +40,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -128,7 +129,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/2p/fracture/main.cc b/test/porousmediumflow/2p/fracture/main.cc
index cdf25936e50561cbd277cd28c5251d916523fe51..e86f55d43a776c4ac6b07da75be6b5b56b2a3c5a 100644
--- a/test/porousmediumflow/2p/fracture/main.cc
+++ b/test/porousmediumflow/2p/fracture/main.cc
@@ -35,7 +35,8 @@
 #include <dumux/common/dumuxmessage.hh>
 #include <dumux/io/container.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 #include <dumux/assembly/fvassembler.hh>
@@ -130,7 +131,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/2p2c/chemicalnonequilibrium/main.cc b/test/porousmediumflow/2p2c/chemicalnonequilibrium/main.cc
index 3f610a12581bd30d0a66ee1ecee5e0f5548827b3..3f922025d33feabd2bd98ecf98efc4a4a4742715 100644
--- a/test/porousmediumflow/2p2c/chemicalnonequilibrium/main.cc
+++ b/test/porousmediumflow/2p2c/chemicalnonequilibrium/main.cc
@@ -33,7 +33,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/porousmediumflow/nonequilibrium/newtonsolver.hh>
 
@@ -122,7 +123,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/2p2c/evaporation/main.cc b/test/porousmediumflow/2p2c/evaporation/main.cc
index f5073314d37f3a55e5168e6fd9890f8f08ea7d42..c018c1b9b51cc1a23432c046775b9a96e5ab386c 100644
--- a/test/porousmediumflow/2p2c/evaporation/main.cc
+++ b/test/porousmediumflow/2p2c/evaporation/main.cc
@@ -33,7 +33,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -113,7 +114,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/2p2c/mpnccomparison/main.cc b/test/porousmediumflow/2p2c/mpnccomparison/main.cc
index c341055be2563781580fbc90e7f44f5c730357c5..01045dbade396db35296f0e90162ff84e32f2e7c 100644
--- a/test/porousmediumflow/2p2c/mpnccomparison/main.cc
+++ b/test/porousmediumflow/2p2c/mpnccomparison/main.cc
@@ -33,7 +33,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -118,7 +119,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/2pnc/diffusion/main.cc b/test/porousmediumflow/2pnc/diffusion/main.cc
index 4cd4e4d8498d59436c0b0edd0c342f8dbc8ec0be..484210ce02a808610832b9edc9ee3f57c63269e4 100644
--- a/test/porousmediumflow/2pnc/diffusion/main.cc
+++ b/test/porousmediumflow/2pnc/diffusion/main.cc
@@ -34,7 +34,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -116,7 +117,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver =  std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/2pnc/fuelcell/main.cc b/test/porousmediumflow/2pnc/fuelcell/main.cc
index f48a35389648e46c796b03dd8f0191a491a835fb..974777b5e7bc009cc8b5794811964942c57e56e7 100644
--- a/test/porousmediumflow/2pnc/fuelcell/main.cc
+++ b/test/porousmediumflow/2pnc/fuelcell/main.cc
@@ -33,7 +33,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -116,7 +117,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/2pnc/surfactant/main.cc b/test/porousmediumflow/2pnc/surfactant/main.cc
index 2e49cf2a42491aa55db3296361cc08052e4f8176..7571069f60f6c92b3fedc01b685ba42ab8c59dff 100644
--- a/test/porousmediumflow/2pnc/surfactant/main.cc
+++ b/test/porousmediumflow/2pnc/surfactant/main.cc
@@ -33,7 +33,8 @@
 #include <dumux/io/grid/gridmanager_yasp.hh>
 
 #include <dumux/linear/linearsolvertraits.hh>
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 
 #include "properties.hh"
 
@@ -100,7 +101,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/2pncmin/isothermal/main.cc b/test/porousmediumflow/2pncmin/isothermal/main.cc
index 547eb747aeb46c897a530a5a2969e9fe60360614..a15d6a586f3529021e07952ef25be5a2967301e1 100644
--- a/test/porousmediumflow/2pncmin/isothermal/main.cc
+++ b/test/porousmediumflow/2pncmin/isothermal/main.cc
@@ -33,7 +33,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -135,7 +136,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/2pncmin/nonisothermal/main.cc b/test/porousmediumflow/2pncmin/nonisothermal/main.cc
index 25219dd2a68ae7f7464b9ebc2cc587b235ba8cae..be9bb5bf9a105283da0171410c553c743581ba09 100644
--- a/test/porousmediumflow/2pncmin/nonisothermal/main.cc
+++ b/test/porousmediumflow/2pncmin/nonisothermal/main.cc
@@ -33,7 +33,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -135,7 +136,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/3p/convection/main.cc b/test/porousmediumflow/3p/convection/main.cc
index c52f2b757835e28536aeab54be1f0c60929acdd9..4f9ec2f8ac4fa356025aaa36d170dd87deed770f 100644
--- a/test/porousmediumflow/3p/convection/main.cc
+++ b/test/porousmediumflow/3p/convection/main.cc
@@ -32,7 +32,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -120,7 +121,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/3p/infiltration/main.cc b/test/porousmediumflow/3p/infiltration/main.cc
index f072bcecbce259cd4581b431bce39a6c556f6630..e6a8f183ba3c7a6d24fe28e461328eafe0fa1200 100644
--- a/test/porousmediumflow/3p/infiltration/main.cc
+++ b/test/porousmediumflow/3p/infiltration/main.cc
@@ -35,7 +35,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -118,7 +119,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/3p3c/infiltration/main.cc b/test/porousmediumflow/3p3c/infiltration/main.cc
index cc497ba9e3c6d5cecfde981ed7b75f2ded485863..ea8bfe4ab461438416b5ad42e0f8088fa85aa61b 100644
--- a/test/porousmediumflow/3p3c/infiltration/main.cc
+++ b/test/porousmediumflow/3p3c/infiltration/main.cc
@@ -34,7 +34,8 @@
 #include <dumux/common/initialize.hh>
 #include <dumux/common/timeloop.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -124,7 +125,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/3p3c/kuevette/main.cc b/test/porousmediumflow/3p3c/kuevette/main.cc
index 78c02e7af7a0e904262906ac63aaef201e8885ff..389544bc9c1496532ba95bd6dbb7ababf3eaeab6 100644
--- a/test/porousmediumflow/3p3c/kuevette/main.cc
+++ b/test/porousmediumflow/3p3c/kuevette/main.cc
@@ -34,7 +34,8 @@
 #include <dumux/common/dumuxmessage.hh>
 #include <dumux/common/timeloop.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -120,7 +121,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/co2/main.cc b/test/porousmediumflow/co2/main.cc
index 5a8df39be24fefce49e800438e2a62c13ff310b7..4e75642d591183b7f0beb1cdc51f793abbed5f7f 100644
--- a/test/porousmediumflow/co2/main.cc
+++ b/test/porousmediumflow/co2/main.cc
@@ -33,7 +33,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -120,7 +121,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/mpnc/2p2ccomparison/main.cc b/test/porousmediumflow/mpnc/2p2ccomparison/main.cc
index 6d32570198d36a37fa22804be43da493f86e1d09..5f28f88ac6ed3f882c87f570ebdbfda6bb28a28e 100644
--- a/test/porousmediumflow/mpnc/2p2ccomparison/main.cc
+++ b/test/porousmediumflow/mpnc/2p2ccomparison/main.cc
@@ -33,7 +33,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -118,7 +119,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/mpnc/kinetic/main.cc b/test/porousmediumflow/mpnc/kinetic/main.cc
index 28b65d70941aa98681b57a4cf4474f756077a55b..0776d60d7b690f21d65c9821b94675fafcb9cc44 100644
--- a/test/porousmediumflow/mpnc/kinetic/main.cc
+++ b/test/porousmediumflow/mpnc/kinetic/main.cc
@@ -33,7 +33,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/porousmediumflow/nonequilibrium/newtonsolver.hh>
 
@@ -124,7 +125,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/mpnc/obstacle/main.cc b/test/porousmediumflow/mpnc/obstacle/main.cc
index 6d32570198d36a37fa22804be43da493f86e1d09..5f28f88ac6ed3f882c87f570ebdbfda6bb28a28e 100644
--- a/test/porousmediumflow/mpnc/obstacle/main.cc
+++ b/test/porousmediumflow/mpnc/obstacle/main.cc
@@ -33,7 +33,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -118,7 +119,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/solidenergy/main.cc b/test/porousmediumflow/solidenergy/main.cc
index af34844b23289632c5fb9005faa673e7b14ff6eb..4395c638dcc8cdf3863646137f0f2404df978bce 100644
--- a/test/porousmediumflow/solidenergy/main.cc
+++ b/test/porousmediumflow/solidenergy/main.cc
@@ -33,7 +33,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 
@@ -117,7 +118,8 @@ int main(int argc, char** argv)
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, xOld);
 
     // the linear solver
-    using LinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using LinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                               LinearAlgebraTraitsFromAssembler<Assembler>>;
     auto linearSolver = std::make_shared<LinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
diff --git a/test/porousmediumflow/tracer/2ptracer/main.cc b/test/porousmediumflow/tracer/2ptracer/main.cc
index f1633ed7f1095e53742b6cb9766e365fdbd58c9e..d496c20f15c5ab5fb99cec6b50f744cb087cdf03 100644
--- a/test/porousmediumflow/tracer/2ptracer/main.cc
+++ b/test/porousmediumflow/tracer/2ptracer/main.cc
@@ -34,7 +34,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/dumuxmessage.hh>
 
-#include <dumux/linear/amgbackend.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/linear/seqsolverbackend.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
@@ -131,7 +132,8 @@ int main(int argc, char** argv)
     auto twoPAssembler = std::make_shared<TwoPAssembler>(twoPProblem, gridGeometry, twoPGridVariables, timeLoop, pOld);
 
     // the linear solver
-    using TwoPLinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
+    using TwoPLinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                                   LinearAlgebraTraitsFromAssembler<TwoPAssembler>>;
     auto twoPLinearSolver = std::make_shared<TwoPLinearSolver>(leafGridView, gridGeometry->dofMapper());
 
     // the non-linear solver
@@ -162,10 +164,6 @@ int main(int argc, char** argv)
     auto tracerGridVariables = std::make_shared<TracerGridVariables>(tracerProblem, gridGeometry);
     tracerGridVariables->init(x);
 
-    // the linear solver
-    using TracerLinearSolver = AMGBiCGSTABBackend<LinearSolverTraits<GridGeometry>>;
-    auto tracerLinearSolver = std::make_shared<TracerLinearSolver>(leafGridView, gridGeometry->dofMapper());
-
      //! the linear system
     using JacobianMatrix = GetPropType<TracerTypeTag, Properties::JacobianMatrix>;
     auto A = std::make_shared<JacobianMatrix>();
@@ -176,6 +174,11 @@ int main(int argc, char** argv)
     auto tracerAssembler = std::make_shared<TracerAssembler>(tracerProblem, gridGeometry, tracerGridVariables, timeLoop, xOld);
     tracerAssembler->setLinearSystem(A, r);
 
+    // the linear solver
+    using TracerLinearSolver = AMGBiCGSTABIstlSolver<LinearSolverTraits<GridGeometry>,
+                                                     LinearAlgebraTraitsFromAssembler<TracerAssembler>>;
+    auto tracerLinearSolver = std::make_shared<TracerLinearSolver>(leafGridView, gridGeometry->dofMapper());
+
     // set the flux, density and saturation from the 2p problem
     tracerProblem->spatialParams().setVolumeFlux(volumeFlux_);
     tracerProblem->spatialParams().setDensity(density_);