What are the different possible approaches to system error correction in compilers?

Compilers do not correct errors let alone system errors. A system error is a systematic failure, either within the hardware itself or in its software configuration. A compiler is specifically concerned with program source code and whether or not it is capable of being compiled into object code. If not, the compiler will tell you where the problem lies, but you still have to fix it yourself. It will not detect system errors of any kind.