A clear comparison between FEM and FDM can not be done referring to a 25 year old dokument. Stability for higher order FDM started 1994 with a work by Mark H. Carpenter (in Journal of Computational Physics 111 (2), 220-236, 1994) , showing how to impose BC in a provably stable way. These FDM are based on SBP, combined with SAT treatment of BC, leading to an ODE system (as Magnus Svärd mentioned earlier). 

Since 1994 much have happened and the maturity of FDM in the framework of SBP-SAT, in particular for wave-dominated problems have surpassed FEM (unless the geometry is extremely complex). Search for SBP and SAT in google and you will notice that this is now a well-spread methodology for solving time-dependent PDE.

For parabolic problems FEM is a good pick, especially if the geometry is complex. If the geometry is not too complex I would say FDM SBP-SAT is not too difficult to implement and would most likely lead to a faster code (given some chosen error tolerance). I have a paper from 2012 (in Journal of Scientific Computing 51 (3), 650-682, 2012) constructing suitable SBP operators for parabolic (and second order hyperbolic) problem.

As for time-integration. Once you have discretised the parabolic PDE in space (using either FEM or SBP-SAT FDM) use a standard implicit solver like Euler backward or use ODE15s in Matlab (if you are familiar with that). There is no need for accurate time-integration methods here. An easier approach is to us for example the explicit Heuns method in time, but that is most likely less efficient here. For wave-dominated problems the classical 4th order Runge-Kutta method is hard to beat (referring to simplicity and efficiency). 

Good luck with solving the problem!

Best regards Ken

Both FDM and FEM can provide time and space integration. The main advantage of FEM (and Finite Volume Method{FVM}) is its stability. Numerical derivatives are very sensitive to the distance between points. FEM tries to get around this problem by trying to solve a different problem that yields the same answer.

Thank you  Matt for useful information.

which  interpolation function is accurate for galerkin FEM?

One of the main advantages of FEM is its capability to handle many different types of boundary conditions. Also, I refer you to the book "Allaire, P.E., (1985), “Basics of the Finite Element Method: Solid Mechanics, Heat Transfer and Fluid Mechanics”, Wm. C.

Brown Publishers, Iowa."

thank you Amr

Another advantage of FEM is that you can use arbitrary geometries and grids with no need to think about how to set up the discretization of the derivatives.

As for what interpolation function to use, that comes down to what information you want to extract, and how many nodes you would like per element. Cubic should be able to handle whatever you throw at it, but for simplicity you may want to start with linear elements. 

I see a number of unclear claims here. FIrst of all, one should not use ad hoc FDM methods but mathematically sensible methods, i.e., energy stable summation-by-parts (SBP) schemes. These methods handle the same kind of boundary conditions as FEM. They are actually more automatic in the handling of bcs than FEM generally is. Furthermore, they have the same stability properties. Once proper FEM and FDM schemes are considered one can do a fair comparison. 1) Grid generation is usually more automatic for FEM, although not completely for complex geometries. For not so complicated geometries it is also straightforward to generate multi-block structured grids. 2) FDM schemes are generally faster for many reasons and quicker to code. 3) Convection dominated problems are usually not so good with FEM. 

As for space-time integration, it is often convenient (and theoretically easier) to discretize in space first which gives you a system of ODEs. Then you pick your favourite ODE solver and march in time.

In your case, both FEM and FDM should work just fine. 

Dear Dr. Qudsiya Irum

A clear comparison between FDM and FEM can be found in P419 in the following reference:

Weiyan T (1992) Shallow water hydrodynamics: mathematical theory and numerical solution for two-dimensional systems of shallow water equations. Elsevier, Amsterdam

A clear comparison between FEM and FDM can not be done referring to a 25 year old dokument. Stability for higher order FDM started 1994 with a work by Mark H. Carpenter (in Journal of Computational Physics 111 (2), 220-236, 1994) , showing how to impose BC in a provably stable way. These FDM are based on SBP, combined with SAT treatment of BC, leading to an ODE system (as Magnus Svärd mentioned earlier). 

Since 1994 much have happened and the maturity of FDM in the framework of SBP-SAT, in particular for wave-dominated problems have surpassed FEM (unless the geometry is extremely complex). Search for SBP and SAT in google and you will notice that this is now a well-spread methodology for solving time-dependent PDE.

For parabolic problems FEM is a good pick, especially if the geometry is complex. If the geometry is not too complex I would say FDM SBP-SAT is not too difficult to implement and would most likely lead to a faster code (given some chosen error tolerance). I have a paper from 2012 (in Journal of Scientific Computing 51 (3), 650-682, 2012) constructing suitable SBP operators for parabolic (and second order hyperbolic) problem.

As for time-integration. Once you have discretised the parabolic PDE in space (using either FEM or SBP-SAT FDM) use a standard implicit solver like Euler backward or use ODE15s in Matlab (if you are familiar with that). There is no need for accurate time-integration methods here. An easier approach is to us for example the explicit Heuns method in time, but that is most likely less efficient here. For wave-dominated problems the classical 4th order Runge-Kutta method is hard to beat (referring to simplicity and efficiency). 

Good luck with solving the problem!

Best regards Ken

 Thank you ken

Than you Magnus 

I want to combine galerkin technique with FEM

Together with a colleague, we have recently proposed a space-time,

semi-lagrangian technique for parabolic equations

http://epubs.siam.org/doi/abs/10.1137/140969713

https://www.degruyter.com/view/j/caim.2016.7.issue-3/caim-2016-0022/caim-2016-0022.xml

that is easy to implement in either FD or FE setting. You might find it useful.