# How to define a test function for H^(1/2)

**URL:** <https://community.freefem.org/t/how-to-define-a-test-function-for-h-1-2/1401>\
**Category:** General Discussion\
**Created:** [December 22, 2021, 4:10pm UTC](https://community.freefem.org/t/how-to-define-a-test-function-for-h-1-2/1401 "2021-12-22T16:10:03Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![sumantkr](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/sumantkr/32/2681_2.png) [@sumantkr](https://community.freefem.org/u/sumantkr)\
**Post date:** [December 22, 2021, 4:10pm UTC](https://community.freefem.org/t/how-to-define-a-test-function-for-h-1-2/1401/1 "2021-12-22T16:10:03Z")

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Dear all,  
@prj @frederichecht

How to define a test space (trace space) H^(1/2) over an interface of two sub-domains?

or, the only restriction of a test space that is defined for a sub-domain over the interface will work in this case (like in the **Schwarz non-overlapping scheme** )

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**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:** [December 23, 2021, 1:42pm UTC](https://community.freefem.org/t/how-to-define-a-test-function-for-h-1-2/1401/2 "2021-12-23T13:42:13Z")

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This question prove that you do not understand Finite element method.  
Sorry, no answers but Eliseo Chacon Vera [eliseo@um.es](mailto:eliseo@um.es) do some work to approximate H^1/2 norm

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<div class="post-metadata">

**Author:** ![sumantkr](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/sumantkr/32/2681_2.png) [@sumantkr](https://community.freefem.org/u/sumantkr)\
**Post date:** [December 23, 2021, 2:10pm UTC](https://community.freefem.org/t/how-to-define-a-test-function-for-h-1-2/1401/3 "2021-12-23T14:10:17Z")

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Thanks, Prof @frederichecht Sir,  
And sorry for such an elementary question or for a wrong projection.

I am searching for a solution for an elliptic interface problem that includes two sub-domains:

I have to define FE space for \Omega\_1, \Omega\_2 and on interface \Gamma which I did like:

```auto

fespace Vh1(Th1, P1); // for domain1 
fespace Vh2(Th2, P2); // for domain2

Vh1 u1, v1, lambda;
Vh2 u2, v2;

```

for defining a weak form on interface \Gamma I have restricted the weak formulation as:

```auto
int1d(Th1, interface)( lambda * v1 )

```

I am not sure about it;

Kindly comment to sort out this issue.

your comments/suggestions would help me to figure out such interface boundary problems.

or, any relevant source code that can handle such interface boundary problems.

Thanks Sir