THESE NOTES SUMMARISE a course on the finite element solution of Elliptic problems, which took place in August 1978, in Bangalore. I would like to thank Professor Ramanathan without whom this course would not have been possible, and Dr. K. Balagangadharan who welcomed me in Bangalore. Mr. Vijayasundaram wrote these notes and gave them a much better form that what I would have been able to. Finally, I am grateful to all the people I met in Bangalore since they helped me to discover the smile of India and the depth of Indian civilization. Bertrand Mercier Paris, June 7, 1979. 1. SOBOLEV SPACES IN THIS CHAPTER the notion of Sobolev space Hl(n) is introduced. We state the Sobolev imbedding theorem, Rellich theorem, and Trace theorem for Hl(n), without proof. For the proof of the theorems the reader is r~ferred to ADAMS [1]. n 1. 1. NOTATIONS. Let n em (n = 1, ~ or 3) be an open set. Let r denote the boundary of 0, it is lSSlimed to be bounded and smooth. Let 2 2 L (n) = {f: Jlfl dx < ~} and n (f,g) = f fg dx. n Then L2(n) is a Hilbert space with (•,•) as the scalar product.