I have noticed that most finite element textbooks tend to ignore the first derivative. they always consider the undifferentiated/natural term, the second derivative and possible the fourth. in my search, I found out that in Reddy’s solution to Problem 3.2, he multiplied the first derivative by the residual and he did not reduce it. He noted that ” the term involving b(the first derivative) is not integrated by parts because it does not reduce the differentiability required of the approximation functions.” whereas ‘Finite Element Methods for Partial Diﬀerential Equations by J.J.W. van der Vegt and O. Bokhove Faculty of Mathematical Sciences University of Twente’ weakened the first derivative in page 6, chapter two by integrating by part. I would love to know the correct procedure to problems having a first derivative.

Both cases give the weak form since the continuity requirements on the primary variable has been weakened. There is no “correct” solution. What we can look at is which gives the more accurate solution. There are people who have refused to reduce the continuity conditions through integration by parts. Look at the discussion here: http://www.researchgate.net/post/Why_is_it_important_to_have_a_Weak_Formulation_for_FEM
If you are inclined, I think the comparison between the accuracy obtained by retaining the strong form can be looked at in specific cases. I don’t know how much work has been done on this.

In an equation such as ay”+by’+cy=0. I have noticed that most finite element textbooks to ignore or do not involve the first derivative(y’). they always consider the undifferentiated/natural term(y), the second derivative(y”) and possible the fourth(y^iv). In my search, I found out that in Reddy’s solution to Problem 3.2, he multiplied the first derivative(y’) by the residual(w) and he did not reduce it. He noted that ” the term involving b(the first derivative) is not integrated by parts because it does not reduce the differentiability required of the approximation functions.” whereas ‘Finite Element Methods for Partial Diﬀerential Equations by J.J.W. van der Vegt and O. Bokhove Faculty of Mathematical Sciences University of Twente’ weakened the first derivative in page 6, chapter two by integrating by part. I would love to know the correct procedure to problems having a first derivative.

Further to my comment on the above note, I want to further draw your attention to the fact that there are FEA methods that do not even create a weak form at all. Two recent papers on this issue are available to you in the link, http://1drv.ms/1Egv2ss . I think it is a good thing to take a closer look at this and even perhaps do a study on the accuracy and other issues that may be associated with use of strong Finite Element Methods (SFEM) as opposed to weak FEM.

Many thanks for uploading your lecture notes on the open web. I am not a student of the course but I do use your lecture notes as supplement to my notes. In modal analysis, which type of FE will give a good solution of the weak FE and the SFE ?

I am not an expert on this. However, I think you are really looking at the Eigenvalue Problem. I have not seen anything done on it from the Strong FEA people. Virtually all the literature I am aware of is in the Weak Form FEA. There are many examples in textbooks. The text for the current course is recommended.

I have noticed that most finite element textbooks tend to ignore the first derivative. they always consider the undifferentiated/natural term, the second derivative and possible the fourth. in my search, I found out that in Reddy’s solution to Problem 3.2, he multiplied the first derivative by the residual and he did not reduce it. He noted that ” the term involving b(the first derivative) is not integrated by parts because it does not reduce the differentiability required of the approximation functions.” whereas ‘Finite Element Methods for Partial Diﬀerential Equations by J.J.W. van der Vegt and O. Bokhove Faculty of Mathematical Sciences University of Twente’ weakened the first derivative in page 6, chapter two by integrating by part. I would love to know the correct procedure to problems having a first derivative.

Both cases give the weak form since the continuity requirements on the primary variable has been weakened. There is no “correct” solution. What we can look at is which gives the more accurate solution. There are people who have refused to reduce the continuity conditions through integration by parts. Look at the discussion here: http://www.researchgate.net/post/Why_is_it_important_to_have_a_Weak_Formulation_for_FEM

If you are inclined, I think the comparison between the accuracy obtained by retaining the strong form can be looked at in specific cases. I don’t know how much work has been done on this.

In an equation such as ay”+by’+cy=0. I have noticed that most finite element textbooks to ignore or do not involve the first derivative(y’). they always consider the undifferentiated/natural term(y), the second derivative(y”) and possible the fourth(y^iv). In my search, I found out that in Reddy’s solution to Problem 3.2, he multiplied the first derivative(y’) by the residual(w) and he did not reduce it. He noted that ” the term involving b(the first derivative) is not integrated by parts because it does not reduce the differentiability required of the approximation functions.” whereas ‘Finite Element Methods for Partial Diﬀerential Equations by J.J.W. van der Vegt and O. Bokhove Faculty of Mathematical Sciences University of Twente’ weakened the first derivative in page 6, chapter two by integrating by part. I would love to know the correct procedure to problems having a first derivative.

Further to my comment on the above note, I want to further draw your attention to the fact that there are FEA methods that do not even create a weak form at all. Two recent papers on this issue are available to you in the link, http://1drv.ms/1Egv2ss . I think it is a good thing to take a closer look at this and even perhaps do a study on the accuracy and other issues that may be associated with use of strong Finite Element Methods (SFEM) as opposed to weak FEM.

Many thanks for uploading your lecture notes on the open web. I am not a student of the course but I do use your lecture notes as supplement to my notes. In modal analysis, which type of FE will give a good solution of the weak FE and the SFE ?

I am not an expert on this. However, I think you are really looking at the Eigenvalue Problem. I have not seen anything done on it from the Strong FEA people. Virtually all the literature I am aware of is in the Weak Form FEA. There are many examples in textbooks. The text for the current course is recommended.