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documentation/Doxygen/Multigrid/Numerics/ContinuousGalerkin.dox File Reference

Functions

<!-- WE COMMENT THIS OUT. I DON 'T THINK IT 'S READY YET Now we are working on a cell-by-cell basis, let us examine the @f$ 1^{st} @f$ cell. We note that only @f$ \phi_0 @f$ and @f$ \phi_1 @f$ are non-zero here, so our original equation reduces to:\f{eqnarray *}{ u_1 \ \sum_{k=1, 2} \int_{c_k}(\nabla \phi_0, \ \phi_0) \+dx \=\ \int_{c_0}(f, \ \phi_0) dx \f} and we drop the @f$ \forall \phi @f$ requirement since only @f$ \phi_0 @f$ is non-zero here. In effect, this @f$ \int_{c_0}(\nabla \phi_0, \ \phi_0) dx \=\ \int_{c_0}(f, \ \phi_0) dx @f$ will become our matrix element, as we shall see shortly. --> The basis of functions that we use in this example are piecewise linear defined in each cell (which is an interval of width @f$ h @f$)
 

Variables

<!-- WE COMMENT THIS OUT. I DON 'T THINK IT 'S READY YET Now we are working on a cell-by-cell basis, let us examine the @f$ 1^{st} @f$ cell. We note that only @f$ \phi_0 @f$ and @f$ \phi_1 @f$ are non-zero here, so our original equation reduces to:\f{eqnarray *}{ u_1 \ \sum_{k=1, 2} \int_{c_k}(\nabla \phi_0, \ \phi_0) \+dx \=\ \int_{c_0}(f, \ \phi_0) dx \f} and we drop the @f$ \forall \phi @f$ requirement since only @f$ \phi_0 @f$ is non-zero here. In effect, this @f$ \int_{c_0}(\nabla \phi_0, \ \phi_0) dx \=\ \int_{c_0}(f, \ \phi_0) dx @f$ will become our matrix element, as we shall see shortly. --> The basis of functions that we use in this example are piecewise linear functions
 
And from this we can write down f$ nabla phi_i f$
 
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix elements
 
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ j
 
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will be
 
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By construction
 
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse matrix
 
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other words
 
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other they have only local support We can read off the right hand side values
 
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other they have only local support We can read off the right hand side by taking our known right hand side f$ f f$ and integrating against an appropriate test function
 
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other they have only local support We can read off the right hand side by taking our known right hand side f$ f f$ and integrating against an appropriate test phi_i dx f Please excuse the slight abuse of notation here There should probably be a clearer indication that we move from a known
 
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other they have only local support We can read off the right hand side by taking our known right hand side f$ f f$ and integrating against an appropriate test phi_i dx f Please excuse the slight abuse of notation here There should probably be a clearer indication that we move from a continuous f$ f f$ to some discrete f$ f_i f$ We can demonstrate simply It s worth doing
 
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other they have only local support We can read off the right hand side by taking our known right hand side f$ f f$ and integrating against an appropriate test phi_i dx f Please excuse the slight abuse of notation here There should probably be a clearer indication that we move from a continuous f$ f f$ to some discrete f$ f_i f$ We can demonstrate simply It s worth as when we discuss the discontinuous version of this scenario
 
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other they have only local support We can read off the right hand side by taking our known right hand side f$ f f$ and integrating against an appropriate test phi_i dx f Please excuse the slight abuse of notation here There should probably be a clearer indication that we move from a continuous f$ f f$ to some discrete f$ f_i f$ We can demonstrate simply It s worth as when we discuss the discontinuous version of this it will no longer disappear We take our left hand side and discretise it
 
we integrate over each cell and then take the sum across each of the cells We also consider the terms that enter the f$ k {th} @f$ row
 
we integrate over each cell and then take the sum across each of the cells We also consider the terms that enter the f$ so we are integrating with f$ phi_k phi_k dx
 
f$ hat {n_{j_+}} \ = \ -\hat{n_{{n+1}_-}} @f$
 

Function Documentation

◆ cell()

<!-- WE COMMENT THIS OUT. I DON 'T THINK IT 'S READY YET Now we are working on a cell-by-cell basis, let us examine the @f$ 1^{st} @f$ cell. We note that only @f$ \phi_0 @f$ and @f$ \phi_1 @f$ are non-zero here, so our original equation reduces to:\f{eqnarray *}{ u_1 \ \sum_{k=1, 2} \int_{c_k}(\nabla \phi_0, \ \phi_0) \+dx \=\ \int_{c_0}(f, \ \phi_0) dx \f} and we drop the @f$ \forall \phi @f$ requirement since only @f$ \phi_0 @f$ is non-zero here. In effect, this @f$ \int_{c_0}(\nabla \phi_0, \ \phi_0) dx \=\ \int_{c_0}(f, \ \phi_0) dx @f$ will become our matrix element, as we shall see shortly. --> The basis of functions that we use in this example are piecewise linear defined in each cell ( which is an interval of width @f$ h @ f$)

Variable Documentation

◆ be

And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will be

◆ construction

And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By construction

Definition at line 256 of file ContinuousGalerkin.dox.

◆ doing

And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other they have only local support We can read off the right hand side by taking our known right hand side f$ f f$ and integrating against an appropriate test phi_i dx f Please excuse the slight abuse of notation here There should probably be a clearer indication that we move from a continuous f$ f f$ to some discrete f$ f_i f$ We can demonstrate simply It s worth doing

Definition at line 274 of file ContinuousGalerkin.dox.

◆ dx

we integrate over each cell and then take the sum across each of the cells We also consider the terms that enter the f$ so we are integrating with f$ phi_k phi_k dx
Initial value:
= \sum_j &\int_{c_j}& (\nabla u, \ \nabla \phi_k) dx \\
- &\int_{\partial c_j^-}& (\nabla u \cdot \hat{n_{j_-}}, \ \phi_k) dS(c_j^-) \\
- &\int_{\partial c_j^+}& (\nabla u \cdot \hat{n_{j_+}}, \ \phi_k) dS(c_j^+)
\f}
\li @f$ \partial c_j^\pm @f$ denotes the right and left boundaries of the cell, respectively.
\li @f$ \hat{n_{j_\pm}} @f$ denotes the vector that is normal to the cell at the right and left boundaries respectively.
We can afford to lose these last two terms, since they will both appear in consecutive terms of
the sum across f$ j @f$, but with opposite signs
And from this we can write down f$ nabla phi_i f$
And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ j
<!-- WE COMMENT THIS OUT. I DON 'T THINK IT 'S READY YET Now we are working on a cell-by-cell basis, let us examine the @f$ 1^{st} @f$ cell. We note that only @f$ \phi_0 @f$ and @f$ \phi_1 @f$ are non-zero here, so our original equation reduces to:\f{eqnarray *}{ u_1 \ \sum_{k=1, 2} \int_{c_k}(\nabla \phi_0, \ \phi_0) \+dx \=\ \int_{c_0}(f, \ \phi_0) dx \f} and we drop the @f$ \forall \phi @f$ requirement since only @f$ \phi_0 @f$ is non-zero here. In effect, this @f$ \int_{c_0}(\nabla \phi_0, \ \phi_0) dx \=\ \int_{c_0}(f, \ \phi_0) dx @f$ will become our matrix element, as we shall see shortly. --> The basis of functions that we use in this example are piecewise linear defined in each cell(which is an interval of width @f$ h @f$)
we integrate over each cell and then take the sum across each of the cells We also consider the terms that enter the f$ so we are integrating with f$ phi_k phi_k dx
@ at
double f(const tarch::la::Vector< Dimensions, double > &x)

Definition at line 282 of file ContinuousGalerkin.dox.

Referenced by ExaSeis::Derivatives< Shortcuts, num_nodes >::computeDerivatives_x_3D(), exahype2::aderdg::corrector_addCellContributions_loop_AoS(), exahype2::aderdg::corrector_addFluxContributions_body_AoS(), exahype2::aderdg::corrector_addNcpContributions_body_AoS(), exahype2::aderdg::corrector_addRiemannContributions_body_AoS(), exahype2::aderdg::corrector_addRiemannContributions_loop_AoS(), exahype2::aderdg::corrector_addSourceContributions_body_AoS(), exahype2::aderdg::corrector_adjustSolution_computeMaxEigenvalue_body_AoS(), swift2::kernels::legacy::density_kernel(), Refinement::RefineDownToPosition< Shortcuts >::eval(), Refinement::RefinePosition< Shortcuts >::eval(), Refinement::RefineFilterCube< Shortcuts >::eval(), Refinement::RefineBetweenPositions< Shortcuts >::eval(), swift2::kernels::legacy::force_kernel(), swift2::kernels::legacy::forceKernelWithMasking(), exahype2::aderdg::getCoordinates(), exahype2::aderdg::getCoordinatesOnFace(), ContextCurvilinear< Shortcuts, basisSize >::getElementSize(), ContextCartesian< Shortcuts, basisSize >::initUnknownsPatch(), ContextCurvilinear< Shortcuts, basisSize >::initUnknownsPatch(), ContextDiffuse< Shortcuts, basisSize >::initUnknownsPatch(), kernels::aderdg::generic::c::RungeKuttaIntegrator< SolverType >::largest_dx(), main(), exahype2::aderdg::mapToReferenceCoordinates(), kernels::aderdg::generic::c::maxScaledEigenvalue(), ExaSeis::Derivatives< Shortcuts, num_nodes >::metricDerivatives(), applications::exahype2::ccz4::internal::recomputeAuxiliaryVariablesFD4_centralDifferences_LoopBody(), applications::exahype2::ccz4::internal::recomputeAuxiliaryVariablesFD4_leftDifferences_LoopBody(), applications::exahype2::ccz4::internal::recomputeAuxiliaryVariablesFD4_rightDifferences_LoopBody(), exahype2::aderdg::riemann_maxAbsoluteEigenvalue_body_AoS(), exahype2::aderdg::riemann_maxAbsoluteEigenvalue_loop_AoS(), exahype2::aderdg::riemann_setBoundaryState_body_AoS(), exahype2::aderdg::riemann_setBoundaryState_loop_AoS(), kernels::aderdg::generic::c::riemannSolverLinear(), kernels::aderdg::generic::c::riemannSolverNonlinear(), exahype2::aderdg::tests::ADERDGTest::runADERDGStep(), exahype2::aderdg::rusanovNonlinear_body_AoS(), exahype2::aderdg::rusanovNonlinear_loop_AoS(), kernels::aderdg::generic::c::solutionAdjustment(), exahype2::aderdg::spaceTimePredictor_PicardLoop_addContributions_body_AoS(), exahype2::aderdg::spaceTimePredictor_PicardLoop_addFluxContributionsToRhs_body_AoS(), exahype2::aderdg::spaceTimePredictor_PicardLoop_addNcpContributionToRhs_body_AoS(), exahype2::aderdg::spaceTimePredictor_PicardLoop_addSourceContributionToRhs_body_AoS(), exahype2::aderdg::spaceTimePredictor_PicardLoop_loop_AoS(), exahype2::aderdg::tests::ADERDGTest::testAdvection(), exahype2::aderdg::tests::ADERDGTest::testEuler(), and applications::exahype2::swe::parser::NetCDFHelper::transformIndexCDFRangeToArray().

◆ elements

And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix elements

Definition at line 175 of file ContinuousGalerkin.dox.

Referenced by tl::tuple_for_each(), and tl::tuple_transform().

◆ f$

we integrate over each cell and then take the sum across each of the cells We also consider the terms that enter the f$ so we are integrating with f$ phi_k f$

Definition at line 122 of file ContinuousGalerkin.dox.

◆ function

And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other they have only local support We can read off the right hand side by taking our known right hand side f$ f f$ and integrating against an appropriate test function

Definition at line 262 of file ContinuousGalerkin.dox.

◆ functions

<!-- WE COMMENT THIS OUT. I DON'T THINK IT'S READY YET Now we are working on a cell-by-cell basis, let us examine the @f$ 1^{st} @f$ cell. We note that only @f$ \phi_0 @f$ and @f$ \phi_1 @f$ are non-zero here, so our original equation reduces to: \f{eqnarray*}{ u_1 \ \sum_{k=1,2} \int_{c_k} (\nabla \phi_0, \ \phi_0) \ + dx \ = \ \int_{c_0} (f, \ \phi_0) dx \f} and we drop the @f$ \forall \phi @f$ requirement since only @f$ \phi_0 @f$ is non-zero here. In effect, this @f$ \int_{c_0} (\nabla \phi_0, \ \phi_0) dx \ = \ \int_{c_0} (f, \ \phi_0) dx @f$ will become our matrix element, as we shall see shortly. --> The basis of functions that we use in this example are piecewise linear functions

Definition at line 110 of file ContinuousGalerkin.dox.

◆ hat

f$ hat {n_{j_+}} \ = \ -\hat{n_{{n+1}_-}} @f$

Definition at line 295 of file ContinuousGalerkin.dox.

◆ it

And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other they have only local support We can read off the right hand side by taking our known right hand side f$ f f$ and integrating against an appropriate test phi_i dx f Please excuse the slight abuse of notation here There should probably be a clearer indication that we move from a continuous f$ f f$ to some discrete f$ f_i f$ We can demonstrate simply It s worth as when we discuss the discontinuous version of this it will no longer disappear We take our left hand side and discretise it

◆ j

And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ j

Definition at line 176 of file ContinuousGalerkin.dox.

Referenced by examples::exahype2::mgccz4::admconstraints(), applications::exahype2::ccz4::admconstraints(), peano4::datamanagement::VertexMarker::areAdjacentCellsLocal(), peano4::grid::GridTraversalEventGenerator::areFacesAdjacentToParallelDomainBoundary(), peano4::grid::GridTraversalEventGenerator::areVerticesAdjacentToParallelDomainBoundary(), TP_bindding::AuxiliaryCal(), kernels::riemannsolvers::util::averageRiemannInputs(), TP::TwoPunctures::bicgstab(), TP::TwoPunctures::BY_Aijofxyz(), TP::TwoPunctures::BY_KKofxyz(), TP::TwoPunctures::calculate_derivs(), TP::Utilities::chder(), TP::Utilities::chebev(), TP::Utilities::chebft_Extremes(), TP::Utilities::chebft_Zeros(), toolbox::blockstructured::internal::clearHalfOfHaloLayerAoS(), toolbox::blockstructured::clearHaloLayerAoS(), kernels::index::colMajor(), kernels::index::colMajor(), toolbox::multiprecision::compress(), toolbox::multiprecision::compress(), applications::exahype2::swe::adjoint::DimensionalSplitting::compute_numerical_fluxes(), Numerics::computeAbsA(), ExaSeis::Derivatives< Shortcuts, num_nodes >::computeDerivatives_x_3D(), ExaSeis::Derivatives< Shortcuts, num_nodes >::computeDerivatives_y_3D(), ExaSeis::Derivatives< Shortcuts, num_nodes >::computeDerivatives_z_3D(), kernels::aderdg::generic::c::computeGradQ(), kernels::aderdg::generic::c::computeGradQ(), kernels::aderdg::generic::c::computeGradQi(), kernels::aderdg::generic::c::computeGradQi(), kernels::riemannsolvers::util::computeOsherMatrix(), tarch::la::Matrix< Rows, Cols, Scalar >::convertScalar(), exahype2::fv::copyHalfOfHalo(), peano4::grid::GridTraversalEventGenerator::createGenericCellTraversalEvent(), TP::Utilities::d3tensor(), toolbox::multiprecision::decomposeIntoFourVariants(), TP::TwoPunctures::Derivatives_AB3(), examples::exahype2::mgccz4::enforceMGCCZ4constraints(), tarch::la::equals(), tarch::la::equalsReturnIndex(), Refinement::RefineDownToPositionCustomCoordinates< Shortcuts >::eval(), Refinement::trackVelocity< Shortcuts >::eval(), applications::exahype2::CompressibleNavierStokes::NavierStokesSolver::extrapolateHalo(), toolbox::blockstructured::extrapolatePatchSolutionAndProjectExtrapolatedHaloOntoFaces(), TP::TwoPunctures::F_of_v(), FindInterIndex(), DiffuseInterface< Shortcuts, basisSize >::findRelativePosition(), kernels::aderdg::generic::c::generalisedOsherSolomon(), DiffuseInterface< Shortcuts, basisSize >::getAlphaPatch(), SolverInformationADERDG< order >::getDuDx(), ContextCurvilinear< Shortcuts, basisSize >::getElementSize(), toolbox::finiteelements::getElementWiseAssemblyMatrix(), toolbox::finiteelements::getElementWiseAssemblyMatrix(), toolbox::finiteelements::getElementWiseAssemblyMatrix(), toolbox::finiteelements::getElementWiseAssemblyMatrix(), DiffuseInterface< Shortcuts, basisSize >::getMinDistance(), TP::TwoPunctures::Index(), DiffuseInterface< Shortcuts, basisSize >::initCells(), DiffuseInterface< Shortcuts, basisSize >::initLimits(), ContextCartesian< Shortcuts, basisSize >::initUnknownsPatch(), ContextCurvilinear< Shortcuts, basisSize >::initUnknownsPatch(), ContextDiffuse< Shortcuts, basisSize >::initUnknownsPatch(), Interpolation(), TP::TwoPunctures::J_times_dv(), TP::TwoPunctures::JFD_times_dv(), TP::TwoPunctures::LineRelax_al(), TP::TwoPunctures::LineRelax_be(), tarch::la::lu(), tarch::la::lu(), TP::Utilities::maximum2(), TP::Utilities::maximum3(), kernels::aderdg::generic::c::maxScaledEigenvalue(), ExaSeis::Derivatives< Shortcuts, num_nodes >::metricDerivatives(), TP::Utilities::minimum2(), TP::Utilities::minimum3(), tarch::la::modifiedGramSchmidt(), applications::exahype2::ccz4::ncp(), examples::exahype2::mgccz4::ncp(), applications::exahype2::swe::adjoint::NetCDFWriter::NetCDFWriter(), TP::TwoPunctures::Newton(), TP::TwoPunctures::norm_inf(), kernels::idx2::operator()(), kernels::idx3::operator()(), kernels::idx4::operator()(), kernels::idx5::operator()(), kernels::idx6::operator()(), tarch::la::operator==(), convert::input::PeanoTextPatchFileReader::parsePatch(), tarch::plotter::griddata::unstructured::vtk::VTKBinaryFileWriter::CellWriter::plotHexahedron(), tarch::plotter::griddata::unstructured::vtk::VTKBinaryFileWriter::CellWriter::plotQuadrangle(), toolbox::blockstructured::internal::projectInterpolatedFineCellsOnHaloLayer_AoS(), toolbox::blockstructured::projectPatchHaloOntoFaces(), toolbox::blockstructured::projectPatchSolutionOntoFaces(), applications::exahype2::ccz4::Psi4Calc(), TP::TwoPunctures::PunctEvalAtArbitPosition(), TP::TwoPunctures::PunctEvalAtArbitPositionFast(), TP::TwoPunctures::PunctEvalAtArbitPositionFaster(), TP::TwoPunctures::PunctEvalAtArbitPositionFasterLowRes(), TP::TwoPunctures::PunctTaylorExpandAtArbitPosition(), applications::exahype2::swe::parser::NetCDFReader::readVariable2D(), applications::exahype2::swe::parser::NetCDFReader::readVariable2DHyperslab(), applications::exahype2::ccz4::internal::recomputeAuxiliaryVariablesFD4_4thOrder_LoopBody(), applications::exahype2::ccz4::internal::recomputeAuxiliaryVariablesFD4_centralDifferences_LoopBody(), applications::exahype2::ccz4::internal::recomputeAuxiliaryVariablesFD4_leftDifferences_LoopBody(), applications::exahype2::ccz4::internal::recomputeAuxiliaryVariablesFD4_rightDifferences_LoopBody(), toolbox::finiteelements::reconstructStencilFragments(), toolbox::finiteelements::reconstructUniformStencilFragments(), peano4::grid::reduceGridControlEvents(), TP::TwoPunctures::relax(), toolbox::blockstructured::restrictCell_AoS_averaging(), toolbox::blockstructured::restrictCell_AoS_inject(), toolbox::blockstructured::restrictInnerHalfOfHaloLayer_AoS_averaging(), toolbox::blockstructured::restrictInnerHalfOfHaloLayer_AoS_inject(), kernels::idx2::rev(), Numerics::riemannSolver(), loh::riemannSolver::riemannSolver(), kernels::aderdg::generic::c::riemannSolverLinear(), kernels::aderdg::generic::c::riemannSolverNonlinear(), Numerics::right_eigenvectors(), Numerics::right_eigenvectors_inverse(), kernels::index::rowMajor(), kernels::index::rowMajor(), TP::TwoPunctures::Run(), runBenchmarks(), applications::exahype2::swe::adjoint::SWEAdjoint::set_ghost_layer(), TP::TwoPunctures::set_initial_guess(), TP::TwoPunctures::SetMatrix_JFD(), TP_bindding::SOCCZ4Cal(), kernels::aderdg::generic::c::solutionAdjustment(), SolverInformationADERDG< order >::SolverInformationADERDG(), SolverInformationADERDG< order >::SolverInformationADERDG(), applications::exahype2::ccz4::source(), examples::exahype2::mgccz4::source(), DiffuseInterface< Shortcuts, basisSize >::spanPatchAroundCenter(), TP::TwoPunctures::SpecCoef(), applications::exahype2::euler::sphericalaccretion::SSInfall::startTimeStep(), toolbox::finiteelements::stencilProduct(), toolbox::finiteelements::stencilProduct(), toolbox::finiteelements::stencilProduct(), toolbox::finiteelements::stencilProduct(), toolbox::finiteelements::stencilProduct(), toolbox::finiteelements::stencilProduct(), tarch::la::sum(), applications::exahype2::swe::adjoint::SWEAdjoint::SWEAdjoint(), applications::exahype2::ccz4::TestingOutput(), tarch::la::tests::GramSchmidtTest::testModifiedGramSchmidt(), exahype2::fv::tests::InterpolationRestrictionTest::testPiecewiseConstantInterpolationWithTensorProduct2(), TP::TwoPunctures::TestRelax(), applications::exahype2::ccz4::ThetaOutputNCP(), tarch::la::Matrix< Rows, Cols, Scalar >::toPrettyString(), tarch::la::Matrix< Rows, Cols, Scalar >::toString(), toolbox::multiprecision::uncompressMatrix(), toolbox::multiprecision::uncompressVector(), applications::exahype2::swe::adjoint::DimensionalSplitting::update_unknowns_x_sweep(), and applications::exahype2::swe::adjoint::DimensionalSplitting::update_unknowns_y_sweep().

◆ k

we integrate over each cell and then take the sum across each of the cells We also consider the terms that enter the f$ k {th} @f$ row

Definition at line 278 of file ContinuousGalerkin.dox.

Referenced by examples::exahype2::mgccz4::admconstraints(), applications::exahype2::ccz4::admconstraints(), peano4::grid::Spacetree::areAllVerticesNonHanging(), peano4::grid::Spacetree::areAllVerticesRefined(), peano4::grid::Spacetree::areAllVerticesUnrefined(), TP_bindding::AuxiliaryCal(), kernels::riemannsolvers::util::averageRiemannInputs(), TP::TwoPunctures::calculate_derivs(), TP::Utilities::chebft_Extremes(), TP::Utilities::chebft_Zeros(), toolbox::blockstructured::internal::clearHalfOfHaloLayerAoS(), toolbox::blockstructured::clearHaloLayerAoS(), Numerics::computeAbsA(), ExaSeis::Derivatives< Shortcuts, num_nodes >::computeDerivatives_x_3D(), ExaSeis::Derivatives< Shortcuts, num_nodes >::computeDerivatives_y_3D(), ExaSeis::Derivatives< Shortcuts, num_nodes >::computeDerivatives_z_3D(), toolbox::blockstructured::computeGradient(), toolbox::blockstructured::computeGradientAndReturnMaxDifference(), kernels::aderdg::generic::c::computeGradQ(), kernels::aderdg::generic::c::computeGradQ(), kernels::aderdg::generic::c::computeGradQi(), kernels::aderdg::generic::c::computeGradQi(), kernels::aderdg::generic::c::computeLoehner(), kernels::riemannsolvers::util::computeOsherMatrix(), exahype2::fv::copyHalfOfHalo(), toolbox::particles::createEquallySpacedParticles(), peano4::grid::GridTraversalEventGenerator::createGenericCellTraversalEvent(), toolbox::particles::createParticlesAlignedWithGlobalCartesianMesh(), TP::TwoPunctures::Derivatives_AB3(), peano4::grid::Spacetree::descend(), Refinement::RefineDownToPositionCustomCoordinates< Shortcuts >::eval(), Refinement::trackVelocity< Shortcuts >::eval(), toolbox::blockstructured::extrapolatePatchSolutionAndProjectExtrapolatedHaloOntoFaces(), TP::TwoPunctures::F_of_v(), FindInterIndex(), TP::Utilities::fourft(), kernels::aderdg::generic::c::generalisedOsherSolomon(), toolbox::particles::getAdjacentCellsOwningParticle(), peano4::grid::Spacetree::getAdjacentRanksForNewVertex(), peano4::parallel::Node::getPeriodicBoundaryNumber(), peano4::grid::GridTraversalEventGenerator::getTreeOwningSpacetreeNode(), Numerics::hllem(), peano4::grid::Spacetree::incrementNumberOfAdjacentRefinedLocalCells(), TP::TwoPunctures::Index(), ContextCartesian< Shortcuts, basisSize >::initUnknownsPatch(), ContextCurvilinear< Shortcuts, basisSize >::initUnknownsPatch(), ContextDiffuse< Shortcuts, basisSize >::initUnknownsPatch(), kernels::aderdg::generic::c::RungeKuttaIntegrator< SolverType >::Integrate(), Interpolation(), peano4::grid::Spacetree::isCellSplitCandidate(), peano4::grid::GridTraversalEventGenerator::isSpacetreeNodeLocal(), peano4::grid::isSpacetreeNodeRefined(), TP::TwoPunctures::J_times_dv(), TP::TwoPunctures::JFD_times_dv(), swift2::kernels::legacy::kernelHydro::kernel_deval(), swift2::kernels::legacy::kernelHydro::kernel_eval(), kernels::aderdg::generic::c::RungeKuttaIntegrator< SolverType >::largest_eigenvalue(), TP::TwoPunctures::LineRelax_al(), TP::TwoPunctures::LineRelax_be(), tarch::la::lu(), tarch::la::lu(), exahype2::fv::mapInnerNeighbourVoxelAlongBoundayOntoAuxiliaryVariable(), peano4::grid::Spacetree::markVerticesAroundForkedCell(), TP::Utilities::maximum3(), ExaSeis::Derivatives< Shortcuts, num_nodes >::metricDerivatives(), TP::Utilities::minimum3(), tarch::la::modifiedGramSchmidt(), applications::exahype2::ccz4::ncp(), examples::exahype2::mgccz4::ncp(), kernels::idx3::operator()(), kernels::idx4::operator()(), kernels::idx5::operator()(), kernels::idx6::operator()(), peano4::grid::TraversalVTKPlotter::plotCell(), exahype2::fv::plotPatch(), toolbox::finiteelements::preprocessBoundaryStencil(), toolbox::blockstructured::projectPatchHaloOntoFaces(), toolbox::blockstructured::projectPatchSolutionOntoFaces(), exahype2::fv::internal::projectValueOntoParticle_piecewiseLinear(), applications::exahype2::ccz4::Psi4Calc(), TP::TwoPunctures::PunctEvalAtArbitPosition(), TP::TwoPunctures::PunctEvalAtArbitPositionFast(), TP::TwoPunctures::PunctEvalAtArbitPositionFaster(), TP::TwoPunctures::PunctEvalAtArbitPositionFasterLowRes(), TP::TwoPunctures::PunctTaylorExpandAtArbitPosition(), peano4::grid::Spacetree::receiveAndMergeGridVertexAtHorizontalBoundary(), applications::exahype2::ccz4::internal::recomputeAuxiliaryVariablesFD4_4thOrder_LoopBody(), applications::exahype2::ccz4::internal::recomputeAuxiliaryVariablesFD4_centralDifferences_LoopBody(), applications::exahype2::ccz4::internal::recomputeAuxiliaryVariablesFD4_leftDifferences_LoopBody(), applications::exahype2::ccz4::internal::recomputeAuxiliaryVariablesFD4_rightDifferences_LoopBody(), toolbox::finiteelements::reconstructStencilFragments(), toolbox::finiteelements::reconstructUniformStencilFragments(), exahype2::dg::reduceMaxEigenvalue_patchwise_functors(), TP::TwoPunctures::relax(), loh::riemannSolver::riemannSolver(), kernels::aderdg::generic::c::riemannSolverNonlinear(), Numerics::right_eigenvectors(), Numerics::right_eigenvectors_inverse(), TP::TwoPunctures::Run(), TP::TwoPunctures::set_initial_guess(), TP::TwoPunctures::SetMatrix_JFD(), peano4::utils::setupLookupTableForDLinearised(), applications::exahype2::ccz4::source(), examples::exahype2::mgccz4::source(), TP::TwoPunctures::SpecCoef(), toolbox::finiteelements::stencilProduct(), toolbox::finiteelements::stencilProduct(), toolbox::finiteelements::stencilProduct(), exahype2::fv::rusanov::tests::CopyPatchTest::testCopyPatch(), peano4::grid::tests::GridTraversalEventGeneratorTest::testCreateLeaveCellTraversalEvent1(), applications::exahype2::ccz4::TestingOutput(), tarch::la::tests::GramSchmidtTest::testModifiedGramSchmidt(), applications::exahype2::ccz4::ThetaOutputNCP(), peano4::grid::Spacetree::traverse(), peano4::grid::Spacetree::updateVertexBeforeStore(), peano4::grid::Spacetree::updateVertexRanksWithinCell(), exahype2::fv::validatePatch(), exahype2::aderdg::validatePatch(), and exahype2::aderdg::validateSpacetimePatch().

◆ known

And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other they have only local support We can read off the right hand side by taking our known right hand side f$ f f$ and integrating against an appropriate test phi_i dx f Please excuse the slight abuse of notation here There should probably be a clearer indication that we move from a known

Definition at line 270 of file ContinuousGalerkin.dox.

◆ matrix

◆ scenario

And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other they have only local support We can read off the right hand side by taking our known right hand side f$ f f$ and integrating against an appropriate test phi_i dx f Please excuse the slight abuse of notation here There should probably be a clearer indication that we move from a continuous f$ f f$ to some discrete f$ f_i f$ We can demonstrate simply It s worth as when we discuss the discontinuous version of this scenario

Definition at line 275 of file ContinuousGalerkin.dox.

◆ values

And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other they have only local support We can read off the right hand side values

Definition at line 261 of file ContinuousGalerkin.dox.

Referenced by CCZ4Solver.CCZ4Solver_FV_GlobalAdaptiveTimeStep::__init__(), CCZ4Solver.CCZ4Solver_FV_GlobalAdaptiveTimeStepWithEnclaveTasking::__init__(), CCZ4Solver.CCZ4Solver_FD4_SecondOrderFormulation_GlobalAdaptiveTimeStepWithEnclaveTasking::__init__(), CCZ4Solver.CCZ4Solver_FD4_GlobalAdaptiveTimeStep::__init__(), CCZ4Solver.CCZ4Solver_FD4_GlobalAdaptiveTimeStepWithEnclaveTasking::__init__(), CCZ4Solver.CCZ4Solver_RKDG_GlobalAdaptiveTimeStepWithEnclaveTasking::__init__(), toolbox::blockstructured::clearCell(), ExaSeis::Derivatives< Shortcuts, num_nodes >::computeDerivatives_x_3D(), ExaSeis::Derivatives< Shortcuts, num_nodes >::computeDerivatives_y_3D(), ExaSeis::Derivatives< Shortcuts, num_nodes >::computeDerivatives_z_3D(), tarch::la::DynamicMatrix::DynamicMatrix(), TP::TwoPunctures::F_of_v(), toolbox::multiprecision::findMostAgressiveCompression(), exahype2.solvers.aderdg.kernelgenerator.models.amrRoutinesModel.AMRRoutinesModel::generateCode(), exahype2.solvers.aderdg.kernelgenerator.models.fusedSpaceTimePredictorVolumeIntegralModel.FusedSpaceTimePredictorVolumeIntegralModel::generateCode(), exahype2.solvers.aderdg.kernelgenerator.models.limiterModel.LimiterModel::generateCode(), TP::TwoPunctures::J_times_dv(), TP::TwoPunctures::JFD_times_dv(), TP::TwoPunctures::LinEquations(), tarch::la::Matrix< Rows, Cols, Scalar >::Matrix(), TP::TwoPunctures::NonLinEquations(), tarch::la::DynamicMatrix::operator=(), tarch::la::DynamicMatrix::operator==(), tarch::plotter::pointdata::vtk::VTKWriter::PointDataWriter::plot(), tarch::plotter::griddata::blockstructured::PeanoTextPatchFileWriter::CellDataWriter::plotCell(), tarch::plotter::griddata::unstructured::vtk::VTKBinaryFileWriter::CellDataWriter::plotCell(), tarch::plotter::griddata::unstructured::vtk::VTKTextFileWriter::CellDataWriter::plotCell(), tarch::plotter::griddata::unstructured::vtk::VTUTextFileWriter::CellDataWriter::plotCell(), tarch::plotter::griddata::blockstructured::PeanoTextPatchFileWriter::VertexDataWriter::plotVertex(), tarch::plotter::griddata::unstructured::vtk::VTKBinaryFileWriter::VertexDataWriter::plotVertex(), tarch::plotter::griddata::unstructured::vtk::VTKTextFileWriter::VertexDataWriter::plotVertex(), tarch::plotter::griddata::unstructured::vtk::VTUTextFileWriter::VertexDataWriter::plotVertex(), tarch::plotter::griddata::blockstructured::PeanoTextPatchFileWriter::VertexDataWriter::plotVertex(), TP::TwoPunctures::SetMatrix_JFD(), tarch::la::Vector< Size, Scalar >::Vector(), tarch::la::Vector< Size, Scalar >::Vector(), tarch::la::DynamicMatrix::vectorToString(), and tarch::plotter::griddata::blockstructured::PeanoTextPatchFileWriter::writeMapping().

◆ words

And from this we can write down f$ nabla phi_i nabla phi_i dx but since we are constructing matrix let s investigate the f$ our matrix elements will nabla phi_i dx f By this will be a sparse as these basis functions are chosen to not overlap with each other almost everywhere In other words

Definition at line 257 of file ContinuousGalerkin.dox.