what will happen if i insert the anode of a LED to pin 12 and cathode of the LED to pin 11 and set pin 12 to high and pin 11 to low,will the led glow??
What will happen is you will damage your LED and/or your Arduino since you haven't mentioned a current-limiting resistor (220 ohms could well be suitable).
However the general concept is perfectly sound. You could, for example, have two LEDs, wired in opposite directions. By bringing one pin LOW and the other HIGH, one LED would light up, and by reversing the output to the pins the other LED would light up. If both pins were either HIGH or LOW then neither LED would light up, as there would be no voltage difference between the leads.
There is a calculator for LED resistors.
Yes it will, try it. This is one of the way to use a LED as a light sensor and light source by connecting it to two different Digital pins.