# Inforce homogeneous Neumann BCs using DG FEM **URL:** \ **Category:** General Discussion\ **Created:** [June 24, 2024, 1:24am UTC](https://community.freefem.org/t/inforce-homogeneous-neumann-bcs-using-dg-fem/3339 "2024-06-24T01:24:54Z")\ **Posts on this page:** 4\ **Page:** 1
**Author:** ![noureddine](https://avatars.discourse-cdn.com/v4/letter/n/87869e/32.png) [@noureddine](https://community.freefem.org/u/noureddine)\ **Post date:** [June 24, 2024, 1:24am UTC](https://community.freefem.org/t/inforce-homogeneous-neumann-bcs-using-dg-fem/3339/1 "2024-06-24T01:24:54Z")
Dear experts, I am currently working on implementing the discontinuous Galerkin Finite Element Method (DG-FEM) for solving the Richards equation in a square domain with homogeneous Neumann boundary conditions on the lateral and bottom sides (q.n = 0). Initially, I assumed that these boundary conditions would naturally vanish in the weak form, but this doesn’t seem to be the case. Below, I have outlined the weak form formulations for both the standard Finite Element Method (FEM) and the corresponding DG-FEM: // RE: du/dt = div(Kh grad(h + z)) // Standard FEM: int nn = 50; mesh Th = square(nn, nn, [L_x, L_y]); macro dn(u) (N.x \* dx(u) + N.y \* dy(u) ) // Define the normal derivative varf Richard(h, v, solver = UMFPACK) = int2d(Th, qft = qf1pTlump)( Ah \* h \* v + Kh \* (dx(h) \* dx(v) + dy(h) \* dy(v)) ) - int2d(Th, qft = qf1pTlump)( Ah \* hm \* v - (thetam - thetaold0) \* v - Kh \* dy(v) ) - int1d(Th, 3)(q0 \* v) ; // DG-FEM: problem Richard(h, v, solver = UMFPACK) = int2d(Th, qft = qf1pTlump)( Ah \* h \* v + Kh \* (dy(h) \* dy(v) + dx(h) \* dx(v)) ) + intalledges(Th)( ( jump(v) \* mean(Kh \* dn(h)) - jump(h) \* mean(Kh \* dn(v)) + pena \* jump(h) \* jump(v) ) / nTonEdge ) - int2d(Th, qft = qf1pTlump)( Ah \* hm \* v - (thetanew - thetaold0) \* v - Kh \* dy(v) ) - int1d(Th, 2)(0.15 \* dn(v) + pena \* 0.15 \* v) ; I am encountering issues with the implementation and would greatly appreciate your assistance in resolving them. Any guidance or suggestions you can provide to correct this would be invaluable. Best regards, Noureddine ---
**Author:** ![frederichecht](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/frederichecht/32/15_2.png) [@frederichecht](https://community.freefem.org/u/frederichecht)\ **Post date:** [June 24, 2024, 10:44am UTC](https://community.freefem.org/t/inforce-homogeneous-neumann-bcs-using-dg-fem/3339/2 "2024-06-24T10:44:29Z")
Sorry, DG FEM is not well defined, This a a lot of formulation , so what the formulation. ---
**Author:** ![noureddine](https://avatars.discourse-cdn.com/v4/letter/n/87869e/32.png) [@noureddine](https://community.freefem.org/u/noureddine)\ **Post date:** [June 24, 2024, 1:29pm UTC](https://community.freefem.org/t/inforce-homogeneous-neumann-bcs-using-dg-fem/3339/3 "2024-06-24T13:29:26Z")
/_\_\_\_\_\_\_Continous Backward Euler \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\__/ macro grad(u) [dx(u),dy(u)] // macro dn(u) (N’\*grad(u) ) // def the normal derivative real q0 = 0.5; problem Richard(h, v, solver = GMRES) = int2d(Th, qft = qf1pTlump)( Ah \* h \* v + Kh \* (grad(h)'\*grad(v)) ) - int2d(Th, qft = qf1pTlump)( Ah \* hm \* v - (thetam - thetaold0) \* v - Kh \* dy(v) ) - int1d(Th,3) (dt \* q0 \* v) ; /_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\__/ /_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_Discontinous Galerkin BE \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\__/ problem Richard(h, v, solver = GMRES) = int2d(Th, qft = qf1pTlump)( Ah \* h \* v + Kh \* (grad(h)'\*grad(v)) ) - intalledges(Th)( ( jump(v) \* mean(Kh \* dn(h) ) - pena \* jump(h) \* jump(v) ) / nTonEdge ) - int2d(Th, qft = qf1pTlump)( Ah \* hm \* v - (thetam - thetaold0) \* v - Kh \* dy(v) ) - int1d(Th,3) (dt \* q0 \* v) ; I’m implementing the DG FEM corresponding to the Continuous FEM for the Richards equation. Please note that the Continuous FEM is working well. The flux in the Richards equation is q = -Kh∇(h+y), where h is the unknown. In the DG FEM implementation, I need to ensure that the no-flux (homogeneous Neumann) boundary conditions are correctly applied. Specifically, when setting `intalledges`, I want to ensure it considers all edges except those with no-flux boundary conditions. Can you provide guidance on how to exclude contributions from the edges with no-flux boundary conditions while implementing the DG FEM in FreeFem++? ---
**Author:** ![fb77](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/fb77/32/3796_2.png) [@fb77](https://community.freefem.org/u/fb77)\ **Post date:** [June 24, 2024, 1:44pm UTC](https://community.freefem.org/t/inforce-homogeneous-neumann-bcs-using-dg-fem/3339/4 "2024-06-24T13:44:36Z")
You can just multiply by `(nTonEdge-1)` to delete the boundary edges, see [About intalledges and internal edges - #2 by fb77](https://community.freefem.org/t/about-intalledges-and-internal-edges/3231/2)