I can give the example used in fluid dynamics. The PDE can be rewritten in weak form by using a FEM-like projection over a specific test function, a combination of Heaviside functions, in such a way that you get the integral form of the conservation equation. The interface between two fluids can be considered a common face between two adjacent volume of fluids. A flux function at the interface is required.

Thanks for your reply. Actually , I have some doubts about how to dealt with this interface.

Hi Julio,

I would recommend you to do some reading about domain decomposition methods for PDEs. I suppose your domains are not overlapping, so the balancing domain decomposition (known as BDDC) should be more appropriate.

P.S. I am currently in SJTU, just in case if you would like to have an extended discussion.

Hello Julio Cesar Gonzalez

First, You need to state the process governed by the PDE. For examples Is it steady fluid flow; nonsteady fluid flow; electromagnetism; hydrodynamics; seepage etc

Secondly, you need to know the characteristics of the interface and how it influences the process.

For examples is it a semipermeable or impermeable interface in a fluid fluid flow problem; is it a slit in an elasticity problem; is it a smooth or adhesive interface in elasticity problem.

Based on the interface characteristics you can then plan how to incorporate its influence in the algebraic equations.

Hi Julio,

You are referring to what some call subdomain method. cases of its implementation can be found in the book of Ramon Bargallo for finite element, Lubin for analytical solutons. In electromagnetics its use isnt much different from the usual weak formulation, you need to define an identification of the mesh elements which will allow to incorporate the physical property of a specific region element-wise.

Thank you all of you for your replies in my questions. Well, I am working with the Fick's equation (Parabolic PDE). I want to study the diffusion of a solute in a system composed by two phases and interface between them. Currently, I am having problem with the mass conservation.

I think what matter is the governing equations of ur problem togetter with the properties of the domain which is the solute. Maybe write down a short text of ur problem and its discrete weack formulation. Just to see if everything is well posed?

Hello Julio Cesar

What is the characterics of your interface? Is it fully permeable or semi permeable?

Your problem can be solved in the usual way by FEM. Your domain should be divided into three zones. Zone 1 consists of phase 1 , zone 2 is the interface and zone three is phase 2.

Fick's law in its simplest form takes the form

Ct = DCxx

where C is the concentration.

D can be assigned D1, D2, D3 in the different zones and the system of algebraic equations can be derived using the forward or backward or implicit scheme. The only complication is that the mesh in zones 1 and 3 shall be finely divided to match the thickness of the interface of zone 2.

Dear Julio Cesar,

This depends on the coupling conditions you impose at the interface between the two sub-domains. Let \Omega_k (k = 1, 2) be the two subdomains and \Gamma be the common interface between. For example, if the first condition is that the normal fluxes (\vec{n}_k \cdot \vec{F}_k with \vec{F} being e.g. \vec{Q}_k u_k - D_k \nabla u_k like for advection diffusion) then you take test functions \phi_k \in H^1(\Omega_k) having equal traces at \Gamma. In this case, when you integrate by parts in both \Omega_k and you sum up the resulting, for the boundary integrals along \Gamma you get

\int_\Gamma \phi_1 (\vec{n}_1 \cdot \vec{F}_1) + \phi_2 (\vec{n}_2 \cdot \vec{F}_2) ds,

which vanishes since \vec{n}_1 = - \vec{n}_2 and at \Gamma the two test functions are equal.

The other condition you may have to impose explicitly. E.g. if you want that u_1 and u_2 have equal traces at \Gamma, you state this explicitly.

You may read about this in

Article A linear domain decomposition method for partially saturated...

Another paper involves nonlinear transmission conditions, e.g. r(u_1) = u_2 at \Gamma

Article Analysis and Upscaling of a Reactive Transport Model in Frac...

Good luck,

Sorin

Thank you so much for your replies dears. My system is similar as Amaechi J. Anyaegbunam described. I am working with the classical Fick's law. The characteristic of the interface is not too relevant, the solute just only pass through. The only feature is that the diffusion coefficient at the interface is equal to an average (D1+D2)/2. As Iuliu Sorin Pop said Let \Omega_k (k = 1, 2) be the two subdomains and \Gamma be the common interface between. Initial condition is C(z,t=0) =Co such 'z' belongs to Omega_k (yes, Co is the same for omega 1 and 2). The boundary condition is zero flow at both ends. My question now is how to deal with the interface?? how to do the assembly of the matrix, because i have two dominain and also an interface?? and finally the flux that cross the interface dissapear in myweak formulation??

Hello Julio Cesar

The attached docx file would help solve your problem

thank you so much for submit your comprehensive file. There is a typing mistake in your matrix P, such a matrix must not contain "D". Amaechi J. Anyaegbuna, what do you think about how to deal with the mass conservation?. Actually my formulation is a closed system, the solute scapes from domain 1 to domain 2. The interface condition is D1 Cx=D2Cx, whrre Cx is the partial derivative of C respect the spatial variable x.

Thank you for correcting my mistake Julio.

It can be shown that the interface condition is automatically satisfied when the weak formulation that I presented is implemented. This is because when the weak formulation is derived by the Galerkin procedure an implicit assumption has been made that there is mass conservation. Please, see the attached docx file for proof.

Dear Julio Cesar,

In my answer you can find the explanation about how to incorporate the flux continuity in the weak formulation. It follows automatically. Also, the papers I suggested give the detail.

Alternatively, you can see the two sub-problems (in each sub-domain) as part of a global one (i.e. formulated in the union of the two sub-domains and of the interface), but with jumps in coefficients/nonlinearities. You can proove rigorously the equivalence of both forms.

I would not try to do it by reformulating the matrices, it is simply unnatural and may lead to mistakes.

Good luck,

Sorin

thank you Mr Amaechi J. Anyaegbunam for your effort to send me your weak form. Now I can assure and certify the reliablity of my formulation. Thanks.