Okay have it both ways

.vscode/settings.json

@@ -0,0 +1,6 @@
+{
+    "files.associations": {
+        "ADStokesStressDivergence.C": "cpp",
+        "ADStokesIncompressibility.C": "cpp"
+    }
+}

include/kernels/ADStokesPressure.h

@@ -0,0 +1,25 @@
+//* This file is part of the MOOSE framework
+//* https://www.mooseframework.org
+//*
+//* All rights reserved, see COPYRIGHT for full restrictions
+//* https://github.com/idaholab/moose/blob/master/COPYRIGHT
+//*
+//* Licensed under LGPL 2.1, please see LICENSE for details
+//* https://www.gnu.org/licenses/lgpl-2.1.html
+
+#pragma once
+
+#include "ADVectorKernel.h"
+
+class ADStokesPressure : public ADVectorKernel
+{
+public:
+  static InputParameters validParams();
+
+  ADStokesPressure(const InputParameters & parameters);
+
+protected:
+  virtual ADReal computeQpResidual() override;
+
+  const ADVariableGradient & _grad_pressure;
+};

include/kernels/ADStokesPressureByParts.h

@@ -0,0 +1,25 @@
+//* This file is part of the MOOSE framework
+//* https://www.mooseframework.org
+//*
+//* All rights reserved, see COPYRIGHT for full restrictions
+//* https://github.com/idaholab/moose/blob/master/COPYRIGHT
+//*
+//* Licensed under LGPL 2.1, please see LICENSE for details
+//* https://www.gnu.org/licenses/lgpl-2.1.html
+
+#pragma once
+
+#include "ADVectorKernel.h"
+
+class ADStokesPressureByParts : public ADVectorKernel
+{
+public:
+  static InputParameters validParams();
+
+  ADStokesPressureByParts(const InputParameters & parameters);
+
+protected:
+  virtual ADReal computeQpResidual() override;
+
+  const ADVariableValue & _pressure;
+};

include/kernels/ADStokesStressDivergence.h

@@ -21,6 +21,5 @@ class ADStokesStressDivergence : public ADVectorKernel
 protected:
   virtual ADReal computeQpResidual() override;
 
-  const ADMaterialProperty<RankTwoTensor> & _stress;
-  const ADVariableGradient & _pressure;
+  const ADMaterialProperty<RankTwoTensor> & _deviatoric_stress;
 };

src/kernels/ADStokesPressure.C

@@ -0,0 +1,35 @@
+//* This file is part of the MOOSE framework
+//* https://www.mooseframework.org
+//*
+//* All rights reserved, see COPYRIGHT for full restrictions
+//* https://github.com/idaholab/moose/blob/master/COPYRIGHT
+//*
+//* Licensed under LGPL 2.1, please see LICENSE for details
+//* https://www.gnu.org/licenses/lgpl-2.1.html
+
+#include "ADStokesPressure.h"
+
+registerMooseObject("DeerApp", ADStokesPressure);
+
+InputParameters
+ADStokesPressure::validParams()
+{
+  InputParameters params = ADVectorKernel::validParams();
+  params.addClassDescription("The pressure part of the velocity kernel for Stokes flow, do not integrate by parts");
+
+  params.addRequiredCoupledVar("pressure", "The pressure");
+
+  return params;
+}
+
+ADStokesPressure::ADStokesPressure(const InputParameters & parameters)
+  : ADVectorKernel(parameters),
+    _grad_pressure(adCoupledGradient("pressure"))
+{
+}
+
+ADReal
+ADStokesPressure::computeQpResidual()
+{
+  return _test[_i][_qp] * _grad_pressure[_qp];
+}

src/kernels/ADStokesPressureByParts.C

@@ -0,0 +1,35 @@
+//* This file is part of the MOOSE framework
+//* https://www.mooseframework.org
+//*
+//* All rights reserved, see COPYRIGHT for full restrictions
+//* https://github.com/idaholab/moose/blob/master/COPYRIGHT
+//*
+//* Licensed under LGPL 2.1, please see LICENSE for details
+//* https://www.gnu.org/licenses/lgpl-2.1.html
+
+#include "ADStokesPressureByParts.h"
+
+registerMooseObject("DeerApp", ADStokesPressureByParts);
+
+InputParameters
+ADStokesPressureByParts::validParams()
+{
+  InputParameters params = ADVectorKernel::validParams();
+  params.addClassDescription("The pressure part of the velocity kernel for Stokes flow, do not integrate by parts");
+
+  params.addRequiredCoupledVar("pressure", "The pressure");
+
+  return params;
+}
+
+ADStokesPressureByParts::ADStokesPressureByParts(const InputParameters & parameters)
+  : ADVectorKernel(parameters),
+    _pressure(adCoupledValue("pressure"))
+{
+}
+
+ADReal
+ADStokesPressureByParts::computeQpResidual()
+{
+  return -_grad_test[_i][_qp].tr() * _pressure[_qp];
+}

src/kernels/ADStokesStressDivergence.C

@@ -17,20 +17,17 @@ ADStokesStressDivergence::validParams()
   InputParameters params = ADVectorKernel::validParams();
   params.addClassDescription("The unstabilized stress diveregence kernel for Stokes flow");
 
-  params.addRequiredCoupledVar("pressure", "The pressure");
-
   return params;
 }
 
 ADStokesStressDivergence::ADStokesStressDivergence(const InputParameters & parameters)
   : ADVectorKernel(parameters),
-    _stress(getADMaterialPropertyByName<RankTwoTensor>("stress")),
-    _pressure(adCoupledGradient("pressure"))
+    _deviatoric_stress(getADMaterialPropertyByName<RankTwoTensor>("deviatoric_stress"))
 {
 }
 
 ADReal
 ADStokesStressDivergence::computeQpResidual()
 {
-  return _stress[_qp].contract(_grad_test[_i][_qp]) + _test[_i][_qp] * _pressure[_qp];
+  return _deviatoric_stress[_qp].contract(_grad_test[_i][_qp]);
 }

src/materials/StokesLinearViscous.C

@@ -24,7 +24,7 @@ StokesLinearViscous::validParams()
 
 StokesLinearViscous::StokesLinearViscous(const InputParameters & parameters)
   : DerivativeMaterialInterface<Material>(parameters),
-    _stress(declareADProperty<RankTwoTensor>("stress")),
+    _stress(declareADProperty<RankTwoTensor>("deviatoric_stress")),
     _strain_rate(getADMaterialPropertyByName<RankTwoTensor>("strain_rate")),
     _mu(getADMaterialProperty<Real>("mu"))
 {
@@ -34,4 +34,4 @@ void
 StokesLinearViscous::computeQpProperties()
 {
   _stress[_qp] = 2.0 * _strain_rate[_qp] * _mu[_qp];
-}
+}

src/materials/StokesPowerLawCreep.C

@@ -25,7 +25,7 @@ StokesPowerLawCreep::validParams()
 
 StokesPowerLawCreep::StokesPowerLawCreep(const InputParameters & parameters)
   : DerivativeMaterialInterface<Material>(parameters),
-    _stress(declareADProperty<RankTwoTensor>("stress")),
+    _stress(declareADProperty<RankTwoTensor>("deviatoric_stress")),
     _strain_rate(getADMaterialPropertyByName<RankTwoTensor>("strain_rate")),
     _A(getADMaterialProperty<Real>("A")),
     _n(getADMaterialProperty<Real>("n"))
@@ -41,6 +41,6 @@ StokesPowerLawCreep::computeQpProperties()
 
   auto effective_strain_rate = std::sqrt(2.0 / 3.0 * dev_strain_rate.contract(dev_strain_rate));
   auto effective_stress = std::pow(effective_strain_rate / A, 1.0 / n);
-
+  
   _stress[_qp] = 2.0 / 3.0 * dev_strain_rate / (A * std::pow(effective_stress, n - 1.0));
-}
+}

src/materials/StokesStrainRate.C

@@ -32,4 +32,5 @@ void
 StokesStrainRate::computeQpProperties()
 {
   _strain_rate[_qp] = 0.5 * (_grad_vel[_qp] + _grad_vel[_qp].transpose());
-}
+  _strain_rate[_qp] = _strain_rate[_qp].deviatoric();
+}

tests/stokes/manufactured.i → tests/stokes/patch.i

@@ -77,6 +77,10 @@
   [equil]
     type = ADStokesStressDivergence
     variable = u
+  []
+  [pressure]
+    type = ADStokesPressure
+    variable = u
     pressure = p
   []
   [incompressible]