feat(newton): first newton solver implementation

tests/operators/prepared_gravity_source.cpp

@@ -1,6 +1,7 @@
 #include <catch2/catch_test_macros.hpp>
 #include <catch2/matchers/catch_matchers_floating_point.hpp>
 #include <mfem.hpp>
+#include <stdexcept>
 
 import mean_field;
 import test_helpers;
@@ -9,6 +10,56 @@ using namespace mean_field;
 using Catch::Matchers::WithinAbs;
 namespace prepared_test = gravity_prepared_test_utils;
 
+namespace {
+    [[nodiscard]] mfem::Vector make_folding_displacement(const mean_field::fem::FEM &f) {
+        mfem::ParGridFunction field(f.displacementFes.get());
+        mfem::VectorFunctionCoefficient coefficient(
+            f.mesh->Dimension(), [](const mfem::Vector &position, mfem::Vector &value) {
+                value.SetSize(position.Size());
+                for (int dimension = 0; dimension < position.Size(); ++dimension) {
+                    value(dimension) = -2.0 * position(dimension);
+                }
+            }
+        );
+        field.ProjectCoefficient(coefficient);
+
+        mfem::Vector displacementTrue;
+        field.GetTrueDofs(displacementTrue);
+        return displacementTrue;
+    }
+} // namespace
+
+TEST_CASE(
+    "Prepared Mapped Gravity Source Reports Invalid Candidate Geometry Without Unwinding",
+    tags::gravity_prepared_unit &tags::geometry
+) {
+    auto args  = test_utils::setup_args();
+    fem::FEM f = fem::setup_fem(args.mesh_file, args, 0);
+    REQUIRE(f.okay());
+
+    operators::PreparedMappedGravitySourceOperator preparedOperator(f, *f.domainMapperStateless);
+    mfem::Vector displacement = preparedOperator.GetDisplacementMap().gather(make_folding_displacement(f));
+
+    const auto rejected       = preparedOperator.TryPrepare(displacement);
+    REQUIRE_FALSE(rejected.has_value());
+    CHECK(rejected.error().reason == operators::GravitySourcePreparationRejectionReason::invalid_mapping);
+    CHECK(rejected.error().mappingStatus == mapping::MappingStatus::non_positive_determinant);
+    CHECK_FALSE(preparedOperator.IsPrepared());
+    CHECK_FALSE(preparedOperator.HasVariationData());
+    CHECK(preparedOperator.GetPreparationCount() == 0);
+
+    CHECK_THROWS_AS(preparedOperator.Prepare(displacement), std::domain_error);
+    CHECK_FALSE(preparedOperator.IsPrepared());
+    CHECK(preparedOperator.GetPreparationCount() == 0);
+
+    displacement         = 0.0;
+    const auto recovered = preparedOperator.TryPrepare(displacement);
+    REQUIRE(recovered.has_value());
+    CHECK(preparedOperator.IsPrepared());
+    CHECK(preparedOperator.HasVariationData());
+    CHECK(preparedOperator.GetPreparationCount() == 1);
+}
+
 TEST_CASE(
     "Prepared Mapped Gravity Source Matches Stateless Kernel",
     tags::gravity_prepared