From 0343d62c57f0709717333c5509fb2775c2ff5e40 Mon Sep 17 00:00:00 2001 From: Joseph Signorelli Date: Tue, 28 Oct 2025 16:16:50 -0500 Subject: [PATCH 1/4] Add c0 IP example --- src/examples.md | 19 +++++++++++++++++++ src/img/examples/ex41.png | Bin 0 -> 154355 bytes 2 files changed, 19 insertions(+) create mode 100644 src/img/examples/ex41.png diff --git a/src/examples.md b/src/examples.md index b8520ca9..a6b1d076 100644 --- a/src/examples.md +++ b/src/examples.md @@ -1285,6 +1285,23 @@ We recommend viewing Example 5 and Example 36 before viewing this example._

+
+##Example 41: Clamped Biharmonic + + +This example demonstrates one way that MFEM can be used to solve PDEs with higher-order derivatives by using the $C^0$ interior penalty by [Brenner and Sung](https://doi.org/10.1007/s10915-004-4135-7). The biharmonic equation with clamped boundary conditions, + +$$\nabla^4u=1 \text{ in } \Omega,$$ +$$\nabla u\cdot \hat{n}=0 \text{ on } \partial\Omega,$$ + +is solved using $H^1$-conforming finite elements. Development of custom domain and face integrators are shown. See the example code for a more detailed description of the method. + +_The example has a serial ([ex41.cpp](https://github.com/mfem/mfem/blob/master/examples/ex41.cpp)) +and a parallel ([ex41p.cpp](https://github.com/mfem/mfem/blob/master/examples/ex41p.cpp)) version. +We recommend viewing Example 1 and Example 14 before viewing this example._ +
+
+
##NURBS Example 1: Poisson Problem @@ -2444,6 +2461,8 @@ function update() + showElement("ex38", all1 && all2 && all3 && none) + showElement("ex39", diffusion && h1 && galerkin && all4) + showElement("ex40", darcy && (l2 || hdiv) && (galerkin || mixed) && (gmres || newton)) + + showElement("ex41", all1 && h1 && galerkin && pcg) + // nurbs miniapps + showElement("nurbs_ex1", diffusion && nurbs && h1) diff --git a/src/img/examples/ex41.png b/src/img/examples/ex41.png new file mode 100644 index 0000000000000000000000000000000000000000..97d5f5b98750bd77a33f78382992286bcf9dbd2a GIT binary patch literal 154355 zcmb??1y>tgv^Egjg9LX6Y&bhw1FaiQ+O*Hc^dHY#<+9KJzxx3nOd0Bhf+PZl;xO<-hdZZ8+aXnuo z?`do4?da}Cuj}Y)i=^vgOV2M#|Hj&zo}ZUrh@MwajF(?bh?ibdonBr+H|J1=83~CV zNku_c*FXQLz&@Mg?T?sKowo%+ffuP>sl^c`tnn6uiP@yYENae6ri*V$7TDWM4om7f zoMqNX2kfVj7lkYobTxsMjf|N;03n=70e^8%~O6MMYXZz(!pU_U&GlhAOrRRHXOQNIq{ieR@n@_@}DjoYtE9ETeN{gO%N4^dxpo0 zCQ1PYxC1r-USwR1(jOS9Q{21r77)5HCr8M=hicx(N5b#43+^`~Ir;S9b){rN!P4EJ 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zyFbv#^E|YZVvq`M^_=n7pXldk=Jg{xter)7?AW1mR=L z_nFss-#wAlBqAa=24yVdnp?kqJ^%jZ0UG&&-I6=YkShZ*LX1nCpo*0kzCa{M@Lziq z_?TCi&F29aE$AaYlsKlUv z5qdotc=v6Xv1_i3uDt68&V&4k}uL`1Tbv5;wDAl}F9=nCps z>uAltqQl9=BuqIY^fAV$WBYMupKS4}v*9)X+S=N9^ilt$PZ1IMWE5D)v`|=BNb4Q9 zaQfh73MmWd;h(S>WG)ZV!9UQzHmquwDH3*ps&$XlX8N6aab!}STSP?MG!!D!Mq67O zJAeJVl+koW_la|y1ZM$=mY4C(zxg8o9)9@YD}6)}5gCU93z;@jsT7+Z{8f&;RyTfU zj}iJf%MSo(Y;2^x{lm<;NF*|8&nO}y`DrLbrj3Gv0#@C=l7p|kYw!6HY=ny>xdcG{ z%^_wrhiGVM;Mr$WIj<2Bk*T4Kg-kn!VX%NFeeFH`2PR22Z@q`R?r8;}wY8Py%a>2} zBZ`Pf4$4@_wA0($OJmJy>R8K1d^fY25xI_NC`3d=L?+EdHdPQ25fPaX1r{PAA~IrO%d0000 Date: Tue, 28 Oct 2025 16:19:14 -0500 Subject: [PATCH 2/4] add amg --- src/examples.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/examples.md b/src/examples.md index a6b1d076..9c43484b 100644 --- a/src/examples.md +++ b/src/examples.md @@ -2461,7 +2461,7 @@ function update() + showElement("ex38", all1 && all2 && all3 && none) + showElement("ex39", diffusion && h1 && galerkin && all4) + showElement("ex40", darcy && (l2 || hdiv) && (galerkin || mixed) && (gmres || newton)) - + showElement("ex41", all1 && h1 && galerkin && pcg) + + showElement("ex41", all1 && h1 && galerkin && (pcg || amg)) // nurbs miniapps From 736ad36836bd33ebfcb58d59b93a8f2ede41b3e1 Mon Sep 17 00:00:00 2001 From: Joseph Signorelli Date: Tue, 28 Oct 2025 16:58:47 -0500 Subject: [PATCH 3/4] minor (add "method") --- src/examples.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/examples.md b/src/examples.md index 9c43484b..d6320c92 100644 --- a/src/examples.md +++ b/src/examples.md @@ -1289,7 +1289,7 @@ We recommend viewing Example 5 and Example 36 before viewing this example._ ##Example 41: Clamped Biharmonic -This example demonstrates one way that MFEM can be used to solve PDEs with higher-order derivatives by using the $C^0$ interior penalty by [Brenner and Sung](https://doi.org/10.1007/s10915-004-4135-7). The biharmonic equation with clamped boundary conditions, +This example demonstrates one way that MFEM can be used to solve PDEs with higher-order derivatives by using the $C^0$ interior penalty method by [Brenner and Sung](https://doi.org/10.1007/s10915-004-4135-7). The biharmonic equation with clamped boundary conditions, $$\nabla^4u=1 \text{ in } \Omega,$$ $$\nabla u\cdot \hat{n}=0 \text{ on } \partial\Omega,$$ From 44bde3b93cd3a87bd030250a98d3e44f6594dd52 Mon Sep 17 00:00:00 2001 From: Joseph Signorelli Date: Tue, 28 Oct 2025 17:04:39 -0500 Subject: [PATCH 4/4] Add missing BC --- src/examples.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/examples.md b/src/examples.md index d6320c92..0bf58e76 100644 --- a/src/examples.md +++ b/src/examples.md @@ -1292,7 +1292,7 @@ We recommend viewing Example 5 and Example 36 before viewing this example._ This example demonstrates one way that MFEM can be used to solve PDEs with higher-order derivatives by using the $C^0$ interior penalty method by [Brenner and Sung](https://doi.org/10.1007/s10915-004-4135-7). The biharmonic equation with clamped boundary conditions, $$\nabla^4u=1 \text{ in } \Omega,$$ -$$\nabla u\cdot \hat{n}=0 \text{ on } \partial\Omega,$$ +$$u=0 \text{ and } \nabla u\cdot \hat{n}=0 \text{ on } \partial\Omega,$$ is solved using $H^1$-conforming finite elements. Development of custom domain and face integrators are shown. See the example code for a more detailed description of the method.