I am an undergrad engineering student. I recently completed my second year, with that said, I have taken several calculus courses. Most recently I completed differential equations and multivariable calculus before that. Now I need to study something which uses partial differential equation extensively, it also uses finite element method and optimization techniques. But the issue is I haven't studied such courses yet, PDE course will be next year I think. So what I am looking for is some really good resources on these topics. I need to master these asap (really fast) else I am screwed.
So you wanna "master" the following topics "really fast":
- (Linear) Partial differential equations
- Finite element methods
- Optimization technique in numerical treatments of PDE
Not discouraging you, or keeping you from motivated, but I highly doubt it can be done within say, one or two month or something, assuming you are working towards a summer project that needs some knowledge on these topics.
Based on my experience of numerical PDE, practically speaking, what you really want right now is a crash course (without going too much into theoretical detail) on
How to get a weak formulation of 2D Poisson equation with Dirichlet boundary condition. How to implement finite element method based on the weak formulation.
The reason for this is that the numerical treatment for Poisson equation is really a guideline for designing a much broader type of numerical methods for linear PDE.
Strauss's book covers way too broad topics, it may not be suitable for short term study for a sophomore.
Artem recommended Peter Olver's notes, I checked it, and I think to suit your needs, it suffices to read only Chapter 1 and 11 in his notes.
My recommendation is to do the following things one by one:
The single best introduction on finite element methods is Chapter 0 of The Mathematical Theory of Finite Element Methods by Brenner and Scott, only Chapter 0.
A good introduction on 2D Poisson equation with computation in mind is Chapter 7 in Introduction to Partial Differential Equations: A Computational Approach.
It suffices to know merely what the weak formulation for Poisson equation is, not the extensive Sobolev spaces theory behind it, therefore reading Example 2 in wikipedia's entry is enough for now.
After reading Chapter 0 in Brenner and Scott and knowing what the weak formulation for 2D Poisson equation is, reading Chapter 4 in Numerical Treatment of Partial Differential Equations by Grossmann and Roos is very good for you to understand from theory to programming.
Then get your hands on 50 lines of MATLAB finite element methods. For further technical details of implementation, part III in Theory and Practice of Finite Elements is extremely well organized, though you may not have time or need to read it right now.
And if you have done these reading and hands-on, you will know at least what pde is, what finite element method is doing to solve a pde numerically.
I don't know exactly what "optimization" in your question means though, for there are too many aspects of numerical pde involve optimizing something, a functional, mesh quality, solving a linear equations system, etc. I can update more into my answer if you elaborate this "optimization" you wanna learn.
Peter Olver's Partial Differential Equations is a very nice set of lecture notes (you will need to supplement them with some problems). But hurry, very soon these lecture notes will turn into a book and probably disappear from his site.