+
diff --git a/examples/Makefile.am b/examples/Makefile.am
index 76e265aca15..ed0d6d78e42 100644
--- a/examples/Makefile.am
+++ b/examples/Makefile.am
@@ -74,7 +74,8 @@ SUBDIRS = \
vector_fe/vector_fe_ex6 \
vector_fe/vector_fe_ex7 \
vector_fe/vector_fe_ex8 \
- vector_fe/vector_fe_ex9
+ vector_fe/vector_fe_ex9 \
+ vector_fe/vector_fe_ex10
AUTOMAKE_OPTIONS = subdir-objects
diff --git a/examples/Makefile.in b/examples/Makefile.in
index 1821dd5a68b..409716393e8 100644
--- a/examples/Makefile.in
+++ b/examples/Makefile.in
@@ -662,7 +662,8 @@ SUBDIRS = \
vector_fe/vector_fe_ex6 \
vector_fe/vector_fe_ex7 \
vector_fe/vector_fe_ex8 \
- vector_fe/vector_fe_ex9
+ vector_fe/vector_fe_ex9 \
+ vector_fe/vector_fe_ex10
AUTOMAKE_OPTIONS = subdir-objects
AM_CPPFLAGS = $(libmesh_CPPFLAGS) $(libmesh_optional_INCLUDES) \
diff --git a/examples/vector_fe/vector_fe_ex10/Makefile.am b/examples/vector_fe/vector_fe_ex10/Makefile.am
new file mode 100644
index 00000000000..42dfcf4b244
--- /dev/null
+++ b/examples/vector_fe/vector_fe_ex10/Makefile.am
@@ -0,0 +1,21 @@
+example_name = vector_fe_ex10
+check_SCRIPTS = run.sh
+install_dir = $(examples_install_path)/vector_fe/ex10
+data = grad_div_exact_solution.h solution_function.h vector_fe_ex10.C vector_fe_ex10.in run.sh
+sources = $(data)
+
+CLEANFILES = out.e
+
+# also need to link files for VPATH builds
+if LIBMESH_VPATH_BUILD
+ BUILT_SOURCES = .linkstamp
+.linkstamp:
+ -rm -f vector_fe_ex10.in && $(LN_S) -f $(srcdir)/vector_fe_ex10.in .
+ $(AM_V_GEN)touch .linkstamp
+
+ CLEANFILES += vector_fe_ex10.in .linkstamp
+endif
+
+##############################################
+# include common example environment
+include $(top_srcdir)/examples/Make.common
diff --git a/examples/vector_fe/vector_fe_ex10/Makefile.in b/examples/vector_fe/vector_fe_ex10/Makefile.in
new file mode 100644
index 00000000000..2acc862f031
--- /dev/null
+++ b/examples/vector_fe/vector_fe_ex10/Makefile.in
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+vtkbuild = @vtkbuild@
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+vtkversion = @vtkversion@
+example_name = vector_fe_ex10
+check_SCRIPTS = run.sh
+install_dir = $(examples_install_path)/vector_fe/ex10
+data = grad_div_exact_solution.h solution_function.h vector_fe_ex10.C vector_fe_ex10.in run.sh
+sources = $(data)
+
+######################################################################
+#
+# how to make documentation
+#
+CLEANFILES = out.e $(am__append_1) $(example_name).html stdout.log
+
+# also need to link files for VPATH builds
+@LIBMESH_VPATH_BUILD_TRUE@BUILT_SOURCES = .linkstamp
+
+######################################################################
+#
+# Common environment for all examples
+#
+# This file is included by *every* example. If we are careful, common
+# rules like 'make doc' or 'make run' can be defined here, once and
+# for all.
+#
+AUTOMAKE_OPTIONS = subdir-objects
+AM_CPPFLAGS = $(libmesh_optional_INCLUDES) -I$(top_builddir)/include $(libmesh_contrib_INCLUDES)
+AM_CFLAGS = $(libmesh_CFLAGS)
+AM_CXXFLAGS = $(libmesh_CXXFLAGS)
+AM_LDFLAGS = $(libmesh_LDFLAGS) $(libmesh_contrib_LDFLAGS)
+examples_source_path = $(top_srcdir)/examples
+examples_install_path = $(prefix)/examples
+data_DATA = $(data) $(top_builddir)/contrib/utils/Makefile
+@LIBMESH_DBG_MODE_TRUE@example_dbg_SOURCES = $(sources)
+@LIBMESH_DBG_MODE_TRUE@example_dbg_CPPFLAGS = $(CPPFLAGS_DBG) $(AM_CPPFLAGS)
+@LIBMESH_DBG_MODE_TRUE@example_dbg_CXXFLAGS = $(CXXFLAGS_DBG)
+@LIBMESH_DBG_MODE_TRUE@example_dbg_LDADD = $(top_builddir)/libmesh_dbg.la
+@LIBMESH_DEVEL_MODE_TRUE@example_devel_SOURCES = $(sources)
+@LIBMESH_DEVEL_MODE_TRUE@example_devel_CPPFLAGS = $(CPPFLAGS_DEVEL) $(AM_CPPFLAGS)
+@LIBMESH_DEVEL_MODE_TRUE@example_devel_CXXFLAGS = $(CXXFLAGS_DEVEL)
+@LIBMESH_DEVEL_MODE_TRUE@example_devel_LDADD = $(top_builddir)/libmesh_devel.la
+@LIBMESH_OPT_MODE_TRUE@example_opt_SOURCES = $(sources)
+@LIBMESH_OPT_MODE_TRUE@example_opt_CPPFLAGS = $(CPPFLAGS_OPT) $(AM_CPPFLAGS)
+@LIBMESH_OPT_MODE_TRUE@example_opt_CXXFLAGS = $(CXXFLAGS_OPT)
+@LIBMESH_OPT_MODE_TRUE@example_opt_LDADD = $(top_builddir)/libmesh_opt.la
+@LIBMESH_PROF_MODE_TRUE@example_prof_SOURCES = $(sources)
+@LIBMESH_PROF_MODE_TRUE@example_prof_CPPFLAGS = $(CPPFLAGS_PROF) $(AM_CPPFLAGS)
+@LIBMESH_PROF_MODE_TRUE@example_prof_CXXFLAGS = $(CXXFLAGS_PROF)
+@LIBMESH_PROF_MODE_TRUE@example_prof_LDADD = $(top_builddir)/libmesh_prof.la
+@LIBMESH_OPROF_MODE_TRUE@example_oprof_SOURCES = $(sources)
+@LIBMESH_OPROF_MODE_TRUE@example_oprof_CPPFLAGS = $(CPPFLAGS_OPROF) $(AM_CPPFLAGS)
+@LIBMESH_OPROF_MODE_TRUE@example_oprof_CXXFLAGS = $(CXXFLAGS_OPROF)
+@LIBMESH_OPROF_MODE_TRUE@example_oprof_LDADD = $(top_builddir)/libmesh_oprof.la
+@LIBMESH_DBG_MODE_TRUE@@LIBMESH_DEVEL_MODE_FALSE@@LIBMESH_OPROF_MODE_FALSE@@LIBMESH_OPT_MODE_FALSE@@LIBMESH_PROF_MODE_FALSE@DOC_PROGRAM = example-dbg
+@LIBMESH_DEVEL_MODE_TRUE@@LIBMESH_OPROF_MODE_FALSE@@LIBMESH_OPT_MODE_FALSE@@LIBMESH_PROF_MODE_FALSE@DOC_PROGRAM = example-devel
+@LIBMESH_OPROF_MODE_FALSE@@LIBMESH_OPT_MODE_FALSE@@LIBMESH_PROF_MODE_TRUE@DOC_PROGRAM = example-prof
+@LIBMESH_OPROF_MODE_TRUE@@LIBMESH_OPT_MODE_FALSE@DOC_PROGRAM = example-oprof
+
+######################################################################
+#
+# Choose a flavor to run when making documentation
+#
+@LIBMESH_OPT_MODE_TRUE@DOC_PROGRAM = example-opt
+@LIBMESH_DBG_MODE_TRUE@@LIBMESH_DEVEL_MODE_FALSE@@LIBMESH_OPROF_MODE_FALSE@@LIBMESH_OPT_MODE_FALSE@@LIBMESH_PROF_MODE_FALSE@DOC_METHOD = dbg
+@LIBMESH_DEVEL_MODE_TRUE@@LIBMESH_OPROF_MODE_FALSE@@LIBMESH_OPT_MODE_FALSE@@LIBMESH_PROF_MODE_FALSE@DOC_METHOD = devel
+@LIBMESH_OPROF_MODE_FALSE@@LIBMESH_OPT_MODE_FALSE@@LIBMESH_PROF_MODE_TRUE@DOC_METHOD = prof
+@LIBMESH_OPROF_MODE_TRUE@@LIBMESH_OPT_MODE_FALSE@DOC_METHOD = oprof
+@LIBMESH_OPT_MODE_TRUE@DOC_METHOD = opt
+
+######################################################################
+#
+# Running the tests
+#
+TESTS_ENVIRONMENT = LIBMESH_DIR=$(abs_top_srcdir); export LIBMESH_DIR; \
+ METHODS="$(METHODS)"; export METHODS; petscmajor=@petscmajor@ \
+ petscminor=@petscminor@
+TESTS = $(check_SCRIPTS)
+all: $(BUILT_SOURCES)
+ $(MAKE) $(AM_MAKEFLAGS) all-am
+
+.SUFFIXES:
+.SUFFIXES: .C .lo .o .obj
+$(srcdir)/Makefile.in: @MAINTAINER_MODE_TRUE@ $(srcdir)/Makefile.am $(top_srcdir)/examples/Make.common $(am__configure_deps)
+ @for dep in $?; do \
+ case '$(am__configure_deps)' in \
+ *$$dep*) \
+ ( cd $(top_builddir) && $(MAKE) $(AM_MAKEFLAGS) am--refresh ) \
+ && { if test -f $@; then exit 0; else break; fi; }; \
+ exit 1;; \
+ esac; \
+ done; \
+ echo ' cd $(top_srcdir) && $(AUTOMAKE) --gnu examples/vector_fe/vector_fe_ex10/Makefile'; \
+ $(am__cd) $(top_srcdir) && \
+ $(AUTOMAKE) --gnu examples/vector_fe/vector_fe_ex10/Makefile
+Makefile: $(srcdir)/Makefile.in $(top_builddir)/config.status
+ @case '$?' in \
+ *config.status*) \
+ cd $(top_builddir) && $(MAKE) $(AM_MAKEFLAGS) am--refresh;; \
+ *) \
+ echo ' cd $(top_builddir) && $(SHELL) ./config.status $(subdir)/$@ $(am__maybe_remake_depfiles)'; \
+ cd $(top_builddir) && $(SHELL) ./config.status $(subdir)/$@ $(am__maybe_remake_depfiles);; \
+ esac;
+$(top_srcdir)/examples/Make.common $(am__empty):
+
+$(top_builddir)/config.status: $(top_srcdir)/configure $(CONFIG_STATUS_DEPENDENCIES)
+ cd $(top_builddir) && $(MAKE) $(AM_MAKEFLAGS) am--refresh
+
+$(top_srcdir)/configure: @MAINTAINER_MODE_TRUE@ $(am__configure_deps)
+ cd $(top_builddir) && $(MAKE) $(AM_MAKEFLAGS) am--refresh
+$(ACLOCAL_M4): @MAINTAINER_MODE_TRUE@ $(am__aclocal_m4_deps)
+ cd $(top_builddir) && $(MAKE) $(AM_MAKEFLAGS) am--refresh
+$(am__aclocal_m4_deps):
+
+clean-checkPROGRAMS:
+ @list='$(check_PROGRAMS)'; test -n "$$list" || exit 0; \
+ echo " rm -f" $$list; \
+ rm -f $$list || exit $$?; \
+ test -n "$(EXEEXT)" || exit 0; \
+ list=`for p in $$list; do echo "$$p"; done | sed 's/$(EXEEXT)$$//'`; \
+ echo " rm -f" $$list; \
+ rm -f $$list
+
+example-dbg$(EXEEXT): $(example_dbg_OBJECTS) $(example_dbg_DEPENDENCIES) $(EXTRA_example_dbg_DEPENDENCIES)
+ @rm -f example-dbg$(EXEEXT)
+ $(AM_V_CXXLD)$(example_dbg_LINK) $(example_dbg_OBJECTS) $(example_dbg_LDADD) $(LIBS)
+
+example-devel$(EXEEXT): $(example_devel_OBJECTS) $(example_devel_DEPENDENCIES) $(EXTRA_example_devel_DEPENDENCIES)
+ @rm -f example-devel$(EXEEXT)
+ $(AM_V_CXXLD)$(example_devel_LINK) $(example_devel_OBJECTS) $(example_devel_LDADD) $(LIBS)
+
+example-oprof$(EXEEXT): $(example_oprof_OBJECTS) $(example_oprof_DEPENDENCIES) $(EXTRA_example_oprof_DEPENDENCIES)
+ @rm -f example-oprof$(EXEEXT)
+ $(AM_V_CXXLD)$(example_oprof_LINK) $(example_oprof_OBJECTS) $(example_oprof_LDADD) $(LIBS)
+
+example-opt$(EXEEXT): $(example_opt_OBJECTS) $(example_opt_DEPENDENCIES) $(EXTRA_example_opt_DEPENDENCIES)
+ @rm -f example-opt$(EXEEXT)
+ $(AM_V_CXXLD)$(example_opt_LINK) $(example_opt_OBJECTS) $(example_opt_LDADD) $(LIBS)
+
+example-prof$(EXEEXT): $(example_prof_OBJECTS) $(example_prof_DEPENDENCIES) $(EXTRA_example_prof_DEPENDENCIES)
+ @rm -f example-prof$(EXEEXT)
+ $(AM_V_CXXLD)$(example_prof_LINK) $(example_prof_OBJECTS) $(example_prof_LDADD) $(LIBS)
+
+mostlyclean-compile:
+ -rm -f *.$(OBJEXT)
+
+distclean-compile:
+ -rm -f *.tab.c
+
+@AMDEP_TRUE@@am__include@ @am__quote@./$(DEPDIR)/example_dbg-vector_fe_ex10.Po@am__quote@ # am--include-marker
+@AMDEP_TRUE@@am__include@ @am__quote@./$(DEPDIR)/example_devel-vector_fe_ex10.Po@am__quote@ # am--include-marker
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+ list='$(DISTFILES)'; \
+ dist_files=`for file in $$list; do echo $$file; done | \
+ sed -e "s|^$$srcdirstrip/||;t" \
+ -e "s|^$$topsrcdirstrip/|$(top_builddir)/|;t"`; \
+ case $$dist_files in \
+ */*) $(MKDIR_P) `echo "$$dist_files" | \
+ sed '/\//!d;s|^|$(distdir)/|;s,/[^/]*$$,,' | \
+ sort -u` ;; \
+ esac; \
+ for file in $$dist_files; do \
+ if test -f $$file || test -d $$file; then d=.; else d=$(srcdir); fi; \
+ if test -d $$d/$$file; then \
+ dir=`echo "/$$file" | sed -e 's,/[^/]*$$,,'`; \
+ if test -d "$(distdir)/$$file"; then \
+ find "$(distdir)/$$file" -type d ! -perm -700 -exec chmod u+rwx {} \;; \
+ fi; \
+ if test -d $(srcdir)/$$file && test $$d != $(srcdir); then \
+ cp -fpR $(srcdir)/$$file "$(distdir)$$dir" || exit 1; \
+ find "$(distdir)/$$file" -type d ! -perm -700 -exec chmod u+rwx {} \;; \
+ fi; \
+ cp -fpR $$d/$$file "$(distdir)$$dir" || exit 1; \
+ else \
+ test -f "$(distdir)/$$file" \
+ || cp -p $$d/$$file "$(distdir)/$$file" \
+ || exit 1; \
+ fi; \
+ done
+check-am: all-am
+ $(MAKE) $(AM_MAKEFLAGS) $(check_PROGRAMS) $(check_SCRIPTS)
+ $(MAKE) $(AM_MAKEFLAGS) check-TESTS
+check: $(BUILT_SOURCES)
+ $(MAKE) $(AM_MAKEFLAGS) check-am
+all-am: Makefile $(DATA)
+installdirs:
+ for dir in "$(DESTDIR)$(datadir)"; do \
+ test -z "$$dir" || $(MKDIR_P) "$$dir"; \
+ done
+install: $(BUILT_SOURCES)
+ $(MAKE) $(AM_MAKEFLAGS) install-am
+install-exec: $(BUILT_SOURCES)
+ $(MAKE) $(AM_MAKEFLAGS) install-exec-am
+install-data: install-data-am
+uninstall: uninstall-am
+
+install-am: all-am
+ @$(MAKE) $(AM_MAKEFLAGS) install-exec-am install-data-am
+
+installcheck: installcheck-am
+install-strip:
+ if test -z '$(STRIP)'; then \
+ $(MAKE) $(AM_MAKEFLAGS) INSTALL_PROGRAM="$(INSTALL_STRIP_PROGRAM)" \
+ install_sh_PROGRAM="$(INSTALL_STRIP_PROGRAM)" INSTALL_STRIP_FLAG=-s \
+ install; \
+ else \
+ $(MAKE) $(AM_MAKEFLAGS) INSTALL_PROGRAM="$(INSTALL_STRIP_PROGRAM)" \
+ install_sh_PROGRAM="$(INSTALL_STRIP_PROGRAM)" INSTALL_STRIP_FLAG=-s \
+ "INSTALL_PROGRAM_ENV=STRIPPROG='$(STRIP)'" install; \
+ fi
+mostlyclean-generic:
+
+clean-generic:
+ -test -z "$(CLEANFILES)" || rm -f $(CLEANFILES)
+
+distclean-generic:
+ -test -z "$(CONFIG_CLEAN_FILES)" || rm -f $(CONFIG_CLEAN_FILES)
+ -test . = "$(srcdir)" || test -z "$(CONFIG_CLEAN_VPATH_FILES)" || rm -f $(CONFIG_CLEAN_VPATH_FILES)
+
+maintainer-clean-generic:
+ @echo "This command is intended for maintainers to use"
+ @echo "it deletes files that may require special tools to rebuild."
+ -test -z "$(BUILT_SOURCES)" || rm -f $(BUILT_SOURCES)
+clean: clean-am
+
+clean-am: clean-checkPROGRAMS clean-generic clean-libtool \
+ mostlyclean-am
+
+distclean: distclean-am
+ -rm -f ./$(DEPDIR)/example_dbg-vector_fe_ex10.Po
+ -rm -f ./$(DEPDIR)/example_devel-vector_fe_ex10.Po
+ -rm -f ./$(DEPDIR)/example_oprof-vector_fe_ex10.Po
+ -rm -f ./$(DEPDIR)/example_opt-vector_fe_ex10.Po
+ -rm -f ./$(DEPDIR)/example_prof-vector_fe_ex10.Po
+ -rm -f Makefile
+distclean-am: clean-am distclean-compile distclean-generic \
+ distclean-tags
+
+dvi: dvi-am
+
+dvi-am:
+
+html-am:
+
+info: info-am
+
+info-am:
+
+install-data-am: install-dataDATA
+
+install-dvi: install-dvi-am
+
+install-dvi-am:
+
+install-exec-am:
+
+install-html: install-html-am
+
+install-html-am:
+
+install-info: install-info-am
+
+install-info-am:
+
+install-man:
+
+install-pdf: install-pdf-am
+
+install-pdf-am:
+
+install-ps: install-ps-am
+
+install-ps-am:
+
+installcheck-am:
+
+maintainer-clean: maintainer-clean-am
+ -rm -f ./$(DEPDIR)/example_dbg-vector_fe_ex10.Po
+ -rm -f ./$(DEPDIR)/example_devel-vector_fe_ex10.Po
+ -rm -f ./$(DEPDIR)/example_oprof-vector_fe_ex10.Po
+ -rm -f ./$(DEPDIR)/example_opt-vector_fe_ex10.Po
+ -rm -f ./$(DEPDIR)/example_prof-vector_fe_ex10.Po
+ -rm -f Makefile
+maintainer-clean-am: distclean-am maintainer-clean-generic
+
+mostlyclean: mostlyclean-am
+
+mostlyclean-am: mostlyclean-compile mostlyclean-generic \
+ mostlyclean-libtool
+
+pdf: pdf-am
+
+pdf-am:
+
+ps: ps-am
+
+ps-am:
+
+uninstall-am: uninstall-dataDATA
+
+.MAKE: all check check-am install install-am install-exec \
+ install-strip
+
+.PHONY: CTAGS GTAGS TAGS all all-am am--depfiles check check-TESTS \
+ check-am clean clean-checkPROGRAMS clean-generic clean-libtool \
+ cscopelist-am ctags ctags-am distclean distclean-compile \
+ distclean-generic distclean-libtool distclean-tags distdir dvi \
+ dvi-am html html-am info info-am install install-am \
+ install-data install-data-am install-dataDATA install-dvi \
+ install-dvi-am install-exec install-exec-am install-html \
+ install-html-am install-info install-info-am install-man \
+ install-pdf install-pdf-am install-ps install-ps-am \
+ install-strip installcheck installcheck-am installdirs \
+ maintainer-clean maintainer-clean-generic mostlyclean \
+ mostlyclean-compile mostlyclean-generic mostlyclean-libtool \
+ pdf pdf-am ps ps-am tags tags-am uninstall uninstall-am \
+ uninstall-dataDATA
+
+.PRECIOUS: Makefile
+
+@LIBMESH_VPATH_BUILD_TRUE@.linkstamp:
+@LIBMESH_VPATH_BUILD_TRUE@ -rm -f vector_fe_ex10.in && $(LN_S) -f $(srcdir)/vector_fe_ex10.in .
+@LIBMESH_VPATH_BUILD_TRUE@ $(AM_V_GEN)touch .linkstamp
+
+# Convenience method: 'make run'
+run: $(check_PROGRAMS)
+ LIBMESH_DIR=$(abs_top_srcdir) METHODS="$(METHODS)" $(srcdir)/$(check_SCRIPTS)
+doc: $(example_name).html
+html: $(example_name).html
+$(example_name).html: $(DOC_PROGRAM) \
+ $(top_srcdir)/contrib/bin/ex2html.sh \
+ Makefile
+ $(MAKE) METHODS=$(DOC_METHOD) -s run > stdout.log
+ $(AM_V_GEN) PATH=$(abs_top_srcdir)/contrib/bin:$(PATH) \
+ ex2html.sh $(example_name) $(abs_srcdir)
+
+# Local Variables:
+# mode: makefile
+# End:
+
+##############################################
+# include common example environment
+
+# Tell versions [3.59,3.63) of GNU make to not export all variables.
+# Otherwise a system limit (for SysV at least) may be exceeded.
+.NOEXPORT:
diff --git a/examples/vector_fe/vector_fe_ex10/grad_div_exact_solution.h b/examples/vector_fe/vector_fe_ex10/grad_div_exact_solution.h
new file mode 100644
index 00000000000..48b7bdb1bcb
--- /dev/null
+++ b/examples/vector_fe/vector_fe_ex10/grad_div_exact_solution.h
@@ -0,0 +1,63 @@
+// The libMesh Finite Element Library.
+// Copyright (C) 2002-2024 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
+
+// This library is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+
+// This library is distributed in the hope that it will be useful,
+// but WITHOUT ANY WARRANTY; without even the implied warranty of
+// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+// Lesser General Public License for more details.
+
+// You should have received a copy of the GNU Lesser General Public
+// License along with this library; if not, write to the Free Software
+// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+
+#ifndef GRAD_DIV_EXACT_SOLUTION_H
+#define GRAD_DIV_EXACT_SOLUTION_H
+
+#include "libmesh/libmesh_common.h"
+#include "libmesh/vector_value.h"
+
+using namespace libMesh;
+
+class GradDivExactSolution
+{
+public:
+ GradDivExactSolution() = default;
+ ~GradDivExactSolution() = default;
+
+ RealGradient operator() (Real x, Real y, Real z)
+ {
+ libmesh_ignore(z);
+
+ const Real ux = cos(k*x)*sin(k*y);
+ const Real uy = sin(k*x)*cos(k*y);
+
+ return RealGradient(ux, uy);
+ }
+
+ RealTensor grad(Real x, Real y, Real z)
+ {
+ libmesh_ignore(z);
+
+ const Real dux_dx = -k*sin(k*x)*sin(k*y);
+ const Real dux_dy = k*cos(k*x)*cos(k*y);
+ const Real duy_dx = dux_dy;
+ const Real duy_dy = dux_dx;
+
+ return RealTensor(dux_dx, dux_dy, Real(0), duy_dx, duy_dy);
+ }
+
+ RealGradient forcing(Real x, Real y, Real z)
+ {
+ return (2*k*k + 1)*operator()(x, y, z);
+ }
+
+private:
+ const Real k = pi;
+};
+
+#endif // GRAD_DIV_EXACT_SOLUTION_H
diff --git a/examples/vector_fe/vector_fe_ex10/run.sh b/examples/vector_fe/vector_fe_ex10/run.sh
new file mode 100755
index 00000000000..44a41aa76a0
--- /dev/null
+++ b/examples/vector_fe/vector_fe_ex10/run.sh
@@ -0,0 +1,41 @@
+#!/bin/sh
+
+#set -x
+
+. "$LIBMESH_DIR"/examples/run_common.sh
+
+example_name=vector_fe_ex10
+
+# Note: these problems are particularly ill-conditioned, so we currently
+# resort to a direct solver.
+
+options="dim=2 element_type=TRI6 -pc_type lu"
+run_example "$example_name" "$options"
+
+options="dim=2 element_type=TRI7 -pc_type lu"
+run_example "$example_name" "$options"
+
+options="dim=2 element_type=QUAD8 -pc_type lu"
+run_example "$example_name" "$options"
+
+options="dim=2 element_type=QUAD9 -pc_type lu"
+run_example "$example_name" "$options"
+
+options="dim=2 order=3 element_type=TRI6 -pc_type lu"
+run_example "$example_name" "$options"
+
+options="dim=2 order=3 element_type=TRI7 -pc_type lu"
+run_example "$example_name" "$options"
+
+options="dim=2 order=3 element_type=QUAD8 -pc_type lu"
+run_example "$example_name" "$options"
+
+options="dim=2 order=3 element_type=QUAD9 -pc_type lu"
+run_example "$example_name" "$options"
+
+# Subdividing each hex into 24 tets gets expensive in dbg...
+options="dim=3 element_type=TET14 grid_size=6 -pc_type lu"
+run_example "$example_name" "$options"
+
+options="dim=3 element_type=HEX27 -pc_type lu"
+run_example "$example_name" "$options"
diff --git a/examples/vector_fe/vector_fe_ex10/solution_function.h b/examples/vector_fe/vector_fe_ex10/solution_function.h
new file mode 100644
index 00000000000..1c0b67a6385
--- /dev/null
+++ b/examples/vector_fe/vector_fe_ex10/solution_function.h
@@ -0,0 +1,109 @@
+// The libMesh Finite Element Library.
+// Copyright (C) 2002-2024 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
+
+// This library is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+
+// This library is distributed in the hope that it will be useful,
+// but WITHOUT ANY WARRANTY; without even the implied warranty of
+// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+// Lesser General Public License for more details.
+
+// You should have received a copy of the GNU Lesser General Public
+// License along with this library; if not, write to the Free Software
+// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+
+#ifndef SOLUTION_FUNCTION_H
+#define SOLUTION_FUNCTION_H
+
+// libMesh includes
+#include "libmesh/function_base.h"
+
+// Example includes
+#include "grad_div_exact_solution.h"
+
+// C++ includes
+#include
+
+using namespace libMesh;
+
+class SolutionFunction : public FunctionBase
+{
+public:
+
+ SolutionFunction() = default;
+ ~SolutionFunction() = default;
+
+ virtual Number operator() (const Point &,
+ const Real = 0)
+ { libmesh_not_implemented(); }
+
+ virtual void operator() (const Point & p,
+ const Real,
+ DenseVector & output)
+ {
+ output.zero();
+ const Real x=p(0), y=p(1), z=p(2);
+ output(0) = soln(x, y, z)(0);
+ output(1) = soln(x, y, z)(1);
+ output(2) = soln(x, y, z)(2);
+ }
+
+ virtual Number component(unsigned int component_in,
+ const Point & p,
+ const Real)
+ {
+ DenseVector outvec(3);
+ (*this)(p, 0, outvec);
+ return outvec(component_in);
+ }
+
+ virtual std::unique_ptr> clone() const
+ { return std::make_unique(); }
+
+private:
+
+ GradDivExactSolution soln;
+};
+
+class SolutionGradient : public FunctionBase
+{
+public:
+
+ SolutionGradient() = default;
+ ~SolutionGradient() = default;
+
+ virtual Gradient operator() (const Point &, const Real = 0)
+ { libmesh_not_implemented(); }
+
+ virtual void operator() (const Point & p,
+ const Real,
+ DenseVector & output)
+ {
+ output.zero();
+ const Real x=p(0), y=p(1), z=p(2);
+ output(0) = soln.grad(x, y, z).row(0);
+ output(1) = soln.grad(x, y, z).row(1);
+ output(2) = soln.grad(x, y, z).row(2);
+ }
+
+ virtual Gradient component(unsigned int component_in,
+ const Point & p,
+ const Real)
+ {
+ DenseVector outvec(3);
+ (*this)(p, 0, outvec);
+ return outvec(component_in);
+ }
+
+ virtual std::unique_ptr> clone() const
+ { return std::make_unique(); }
+
+private:
+
+ GradDivExactSolution soln;
+};
+
+#endif // SOLUTION_FUNCTION_H
diff --git a/examples/vector_fe/vector_fe_ex10/vector_fe_ex10.C b/examples/vector_fe/vector_fe_ex10/vector_fe_ex10.C
new file mode 100644
index 00000000000..0badbf647b7
--- /dev/null
+++ b/examples/vector_fe/vector_fe_ex10/vector_fe_ex10.C
@@ -0,0 +1,482 @@
+// The libMesh Finite Element Library.
+// Copyright (C) 2002-2024 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
+
+// This library is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+
+// This library is distributed in the hope that it will be useful,
+// but WITHOUT ANY WARRANTY; without even the implied warranty of
+// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+// Lesser General Public License for more details.
+
+// You should have received a copy of the GNU Lesser General Public
+// License along with this library; if not, write to the Free Software
+// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+
+
+//
Vector Finite Elements Example 10 - Raviart-Thomas elements (grad-div)
+// \author Nuno Nobre
+// \date 2025
+//
+// This example uses Raviart-Thomas elements to solve a model grad-div problem
+// in H(div) in both 2d and 3d: -\nabla (\nabla \cdot \vec{u}) + \vec{u} = f.
+// Note that, unlike the problem in Vector Finite Elements Example 6, this is
+// _not_ an elliptic problem.
+
+// Basic utilities.
+#include "libmesh/string_to_enum.h"
+
+// The solver packages supported by libMesh.
+#include "libmesh/enum_solver_package.h"
+
+// The mesh object and mesh generation and modification utilities.
+#include "libmesh/mesh.h"
+#include "libmesh/mesh_generation.h"
+#include "libmesh/mesh_modification.h"
+
+// Matrix and vector types.
+#include "libmesh/dense_matrix.h"
+#include "libmesh/sparse_matrix.h"
+#include "libmesh/dense_vector.h"
+#include "libmesh/numeric_vector.h"
+
+// The finite element object and the geometric element type.
+#include "libmesh/fe.h"
+#include "libmesh/elem.h"
+
+// Gauss quadrature rules.
+#include "libmesh/quadrature_gauss.h"
+
+// The dof map, which handles degree of freedom indexing.
+#include "libmesh/dof_map.h"
+
+// The system of equations.
+#include "libmesh/equation_systems.h"
+#include "libmesh/linear_implicit_system.h"
+
+// The exact solution and error computation.
+#include "libmesh/exact_solution.h"
+#include "libmesh/enum_norm_type.h"
+#include "solution_function.h"
+
+// I/O utilities.
+#include "libmesh/getpot.h"
+#include "libmesh/exodusII_io.h"
+
+
+// Bring in everything from the libMesh namespace.
+using namespace libMesh;
+
+// Function prototype. This is the function that will assemble
+// the linear system for our grad-div problem. Note that the
+// function will take the EquationSystems object and the
+// name of the system we are assembling as input. From the
+// EquationSystems object we have access to the Mesh and
+// other objects we might need.
+void assemble_graddiv(EquationSystems & es,
+ const std::string & system_name);
+
+int main (int argc, char ** argv)
+{
+ // Initialize libMesh.
+ LibMeshInit init (argc, argv);
+
+ // This example requires a linear solver package.
+ libmesh_example_requires(libMesh::default_solver_package() != INVALID_SOLVER_PACKAGE,
+ "--enable-petsc, --enable-trilinos, or --enable-eigen");
+
+ // Parse the input file.
+ GetPot infile("vector_fe_ex10.in");
+
+ // But allow the command line to override it.
+ infile.parse_command_line(argc, argv);
+
+ // Read in parameters from the command line and the input file.
+ const unsigned int dimension = infile("dim", 2);
+ const unsigned int grid_size = infile("grid_size", 15);
+
+ // Skip higher-dimensional examples on a lower-dimensional libMesh build.
+ libmesh_example_requires(dimension <= LIBMESH_DIM, dimension << "D support");
+
+ // Create a mesh, with dimension to be overridden later, distributed
+ // across the default MPI communicator.
+ Mesh mesh(init.comm());
+
+ // Use the MeshTools::Generation mesh generator to create a uniform
+ // grid on the cube [-1,1]^D. To accomodate Raviart-Thomas elements, we must
+ // use TRI6/7 or QUAD8/9 elements in 2d, or TET14 or HEX27 in 3d.
+ const std::string elem_str = infile("element_type", std::string("TRI6"));
+
+ libmesh_error_msg_if((dimension == 2 && elem_str != "TRI6" && elem_str != "TRI7" && elem_str != "QUAD8" && elem_str != "QUAD9") ||
+ (dimension == 3 && elem_str != "TET14" && elem_str != "HEX27"),
+ "You selected " << elem_str <<
+ " but this example must be run with TRI6, TRI7, QUAD8, or QUAD9 in 2d" <<
+ " or with TET14, or HEX27 in 3d.");
+
+ if (dimension == 2)
+ MeshTools::Generation::build_square (mesh,
+ grid_size,
+ grid_size,
+ -1., 1.,
+ -1., 1.,
+ Utility::string_to_enum(elem_str));
+ else if (dimension == 3)
+ MeshTools::Generation::build_cube (mesh,
+ grid_size,
+ grid_size,
+ grid_size,
+ -1., 1.,
+ -1., 1.,
+ -1., 1.,
+ Utility::string_to_enum(elem_str));
+
+ // Make sure the code is robust against nodal reorderings.
+ MeshTools::Modification::permute_elements(mesh);
+
+ // Print information about the mesh to the screen.
+ mesh.print_info();
+
+ // Create an equation systems object.
+ EquationSystems equation_systems (mesh);
+
+ // Declare the system "GradDiv" and its variable.
+ LinearImplicitSystem & system = equation_systems.add_system("GradDiv");
+
+ // Set the FE approximation order for the vector field variable.
+ const Order vector_order = static_cast(infile("order", 1u));
+
+ libmesh_error_msg_if(vector_order < FIRST || vector_order > ((dimension == 3) ? FIRST : FIFTH),
+ "You selected: " << vector_order <<
+ " but this example must be run with either 1 <= order <= 5 in 2d"
+ " or with order 1 in 3d.");
+
+ // Adds the variable "u" to "GradDiv". "u" will be our vector field.
+ system.add_variable("u", vector_order, RAVIART_THOMAS);
+
+ // Give the system a pointer to the matrix assembly
+ // function. This will be called when needed by the library.
+ system.attach_assemble_function(assemble_graddiv);
+
+ // Initialize the data structures for the equation system.
+ equation_systems.init();
+
+ // Prints information about the system to the screen.
+ equation_systems.print_info();
+
+ // Solve the system "GradDiv". Note that calling this
+ // member will assemble the linear system and invoke
+ // the default numerical solver.
+ system.solve();
+
+ ExactSolution exact_sol(equation_systems);
+
+ SolutionFunction soln_func;
+ SolutionGradient soln_grad;
+
+ // Build FunctionBase* containers to attach to the ExactSolution object.
+ std::vector *> sols(1, &soln_func);
+ std::vector *> grads(1, &soln_grad);
+
+ exact_sol.attach_exact_values(sols);
+ exact_sol.attach_exact_derivs(grads);
+
+ // Use higher quadrature order for more accurate error results.
+ int extra_error_quadrature = infile("extra_error_quadrature", 2);
+ exact_sol.extra_quadrature_order(extra_error_quadrature);
+
+ // Compute the error.
+ exact_sol.compute_error("GradDiv", "u");
+
+ // Print out the error values.
+ libMesh::out << "~~ Vector field (u) ~~"
+ << std::endl;
+ libMesh::out << "L2 error is: "
+ << exact_sol.l2_error("GradDiv", "u")
+ << std::endl;
+ libMesh::out << "HDiv semi-norm error is: "
+ << exact_sol.error_norm("GradDiv", "u", HDIV_SEMINORM)
+ << std::endl;
+ libMesh::out << "HDiv error is: "
+ << exact_sol.hdiv_error("GradDiv", "u")
+ << std::endl;
+
+#ifdef LIBMESH_HAVE_EXODUS_API
+
+ // We write the file in the ExodusII format.
+ ExodusII_IO(mesh).write_equation_systems("out.e", equation_systems);
+
+#endif // #ifdef LIBMESH_HAVE_EXODUS_API
+
+ // All done.
+ return 0;
+}
+
+
+
+// We now define the matrix assembly function for the
+// grad-div system. We need to first compute element
+// matrices and right-hand sides, and then take into
+// account the boundary conditions, which will be handled
+// via a penalty method.
+void assemble_graddiv(EquationSystems & es,
+ const std::string & libmesh_dbg_var(system_name))
+{
+
+ // It is a good idea to make sure we are assembling
+ // the proper system.
+ libmesh_assert_equal_to (system_name, "GradDiv");
+
+ // Get a constant reference to the mesh object.
+ const MeshBase & mesh = es.get_mesh();
+
+ // The dimension that we are running.
+ const unsigned int dim = mesh.mesh_dimension();
+
+ // Get a reference to the LinearImplicitSystem we are solving.
+ LinearImplicitSystem & system = es.get_system("GradDiv");
+
+ // A reference to the DofMap object for this system. The DofMap
+ // object handles the index translation from node and element numbers
+ // to degree of freedom numbers.
+ const DofMap & dof_map = system.get_dof_map();
+
+ // Get a constant reference to the Finite Element type
+ // for the variable in the system.
+ FEType vector_fe_type = dof_map.variable_type(system.variable_number("u"));
+
+ // Build the Finite Element object. Since the
+ // FEBase::build() member dynamically creates memory we will
+ // store the object as a std::unique_ptr. This can be thought
+ // of as a pointer that will clean up after itself. Introduction Example 4
+ // describes some advantages of std::unique_ptr's in the context of
+ // quadrature rules.
+ std::unique_ptr vector_fe (FEVectorBase::build(dim, vector_fe_type));
+
+ // A just-high-enough Gauss quadrature rule for numerical integration.
+ QGauss qrule (dim, vector_fe_type.default_quadrature_order());
+
+ // Tell the finite element object to use our quadrature rule.
+ vector_fe->attach_quadrature_rule (&qrule);
+
+ // Declare a special finite element object for boundary integration.
+ std::unique_ptr vector_fe_face (FEVectorBase::build(dim, vector_fe_type));
+
+ // Boundary integration requires one quadrature rule with dimensionality one
+ // less than the dimensionality of the element.
+ QGauss qface(dim-1, vector_fe_type.default_quadrature_order());
+
+ // Tell the finite element object to use our quadrature rule.
+ vector_fe_face->attach_quadrature_rule (&qface);
+
+ // Here we define some references to cell-specific data that
+ // will be used to assemble the linear system.
+ //
+ // The element Jacobian * quadrature weight at each integration point.
+ const std::vector & JxW = vector_fe->get_JxW();
+
+ // The physical XY locations of the quadrature points on the element.
+ // These might be useful for evaluating spatially varying material
+ // properties at the quadrature points.
+ const std::vector & q_point = vector_fe->get_xyz();
+
+ // The element shape functions evaluated at the quadrature points.
+ const std::vector> & vector_phi = vector_fe->get_phi();
+
+ // The divergence of the element vector shape functions evaluated at the
+ // quadrature points.
+ const std::vector> & div_vector_phi = vector_fe->get_div_phi();
+
+ // Define data structures to contain the element matrix
+ // and right-hand-side vector contribution. Following
+ // basic finite element terminology we will denote these
+ // "Ke" and "Fe". These datatypes are templated on
+ // Number, which allows the same code to work for real
+ // or complex numbers.
+ DenseMatrix Ke;
+ DenseVector Fe;
+
+ // These vectors will hold the degree of freedom indices for
+ // the element. These define where in the global system
+ // the element degrees of freedom get mapped.
+ std::vector dof_indices;
+
+ // The global system matrix
+ SparseMatrix & matrix = system.get_system_matrix();
+
+ // Now we will loop over all the elements in the mesh.
+ // We will compute the element matrix and right-hand-side
+ // contribution.
+ //
+ // Element ranges are a nice way to iterate through all the
+ // elements, or all the elements that have some property. The
+ // range will iterate from the first to the last element on
+ // the local processor.
+ // It is smart to make this one const so that we don't accidentally
+ // mess it up! In case users later modify this program to include
+ // refinement, we will be safe and will only consider the active
+ // elements; hence we use a variant of the
+ // active_local_element_ptr_range.
+ for (const auto & elem : mesh.active_local_element_ptr_range())
+ {
+ // Get the degree of freedom indices for the
+ // current element. These define where in the global
+ // matrix and right-hand-side this element will
+ // contribute to.
+ dof_map.dof_indices (elem, dof_indices);
+
+ // Cache the total number of degrees of freedom on this element,
+ // for use as array and loop bounds later.
+ // We use cast_int to explicitly convert from size() (which may be
+ // 64-bit) to unsigned int (which may be 32-bit but which is definitely
+ // enough to count *local* degrees of freedom.
+ const unsigned int n_dofs =
+ cast_int(dof_indices.size());
+
+ // Compute the element-specific data for the current
+ // element. This involves computing the location of the
+ // quadrature points (q_point) and the shape functions
+ // and their divergences for the current element.
+ vector_fe->reinit (elem);
+
+ // We should also have the same number of degrees of freedom as
+ // shape functions for our variable.
+ libmesh_assert_equal_to (n_dofs, vector_phi.size());
+
+ // Zero the element matrix and right-hand side before
+ // summing them. We use the resize member here because
+ // the number of degrees of freedom might have changed from
+ // the last element. Note that this will be the case if the
+ // element type is different (i.e. the last element was a
+ // triangle, now we are on a quadrilateral).
+
+ // The DenseMatrix::resize() and the DenseVector::resize()
+ // members will automatically zero out the matrix and vector.
+ Ke.resize (n_dofs, n_dofs);
+ Fe.resize (n_dofs);
+
+ // Now loop over the quadrature points. This handles
+ // the numeric integration.
+ for (unsigned int qp=0; qpside_index_range())
+ if (elem->neighbor_ptr(side) == nullptr)
+ {
+ // The value of the shape functions at the quadrature points.
+ const std::vector> & vector_phi_face = vector_fe_face->get_phi();
+
+ // The Jacobian * Quadrature Weight at the quadrature
+ // points on the face.
+ const std::vector & JxW_face = vector_fe_face->get_JxW();
+
+ // The XYZ locations (in physical space) of, and the normals at,
+ // the quadrature points on the face. This is where
+ // we will interpolate the boundary value function.
+ const std::vector & qface_point = vector_fe_face->get_xyz();
+ const std::vector & normals = vector_fe_face->get_normals();
+
+ // Compute the vector shape function values on the element face.
+ vector_fe_face->reinit(elem, side);
+
+ // Some shape functions will be 0 on the face, but for ease of
+ // indexing and generality of code we loop over them anyway.
+ libmesh_assert_equal_to (n_dofs, vector_phi_face.size());
+
+ // Loop over the face quadrature points for integration.
+ for (unsigned int qp=0; qpadd_vector (Fe, dof_indices);
+ }
+
+ // All done!
+}
diff --git a/examples/vector_fe/vector_fe_ex10/vector_fe_ex10.in b/examples/vector_fe/vector_fe_ex10/vector_fe_ex10.in
new file mode 100644
index 00000000000..50d77513f65
--- /dev/null
+++ b/examples/vector_fe/vector_fe_ex10/vector_fe_ex10.in
@@ -0,0 +1,11 @@
+# Solve the 2D or 3D problem
+dim = 2
+
+# The element type
+element_type = TRI6
+
+# The coarse grid size from which to start adaptivity
+grid_size = 15
+
+# Higher quadrature order for more accurate error results
+extra_error_quadrature = 2
diff --git a/examples/vector_fe/vector_fe_ex6/vector_fe_ex6.C b/examples/vector_fe/vector_fe_ex6/vector_fe_ex6.C
index 1a0f75a53a8..2deac6ec80e 100644
--- a/examples/vector_fe/vector_fe_ex6/vector_fe_ex6.C
+++ b/examples/vector_fe/vector_fe_ex6/vector_fe_ex6.C
@@ -16,14 +16,16 @@
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
-//
Vector Finite Elements Example 6 - Raviart-Thomas elements
+//
Vector Finite Elements Example 6 - Raviart-Thomas elements (div-grad)
// \author Nuno Nobre
// \date 2023
//
// This example uses Raviart-Thomas elements to solve a model div-grad problem
-// in H(div) in both 2d and 3d. The problem is simply a div-grad formulation,
-// \vec{u} = -\nabla p, and \nabla \cdot \vec{u} = f, of the Poisson
-// problem in Introduction Example 3, \nabla^2 p = -f.
+// in H(div) in both 2d and 3d. The problem is simply a mixed div-grad
+// formulation, \vec{u} = -\nabla p, and \nabla \cdot \vec{u} = f, of the
+// Poisson problem in Introduction Example 3, \nabla^2 p = -f. In particular,
+// unlike in Introduction Example 3, where we solve solely for the scalar field
+// p, here we solve for both the vector field \vec{u} and the scalar field p.
// Basic utilities.
#include "libmesh/string_to_enum.h"
@@ -151,7 +153,7 @@ int main (int argc, char ** argv)
// Declare the system "DivGrad" and its variables.
LinearImplicitSystem & system = equation_systems.add_system("DivGrad");
- // Set the FE approximation order for the vector and field variables.
+ // Set the FE approximation order for the vector and scalar field variables.
const Order vector_order = static_cast(infile("order", 1u));
const Order scalar_order = static_cast(vector_order - 1u);