How to initialize only few elements of an array with some values?

Here is my trivial approach:

int array[12] = {0};
array[0] = 1; array[4] = 2; array[8] = 3;

However, technically speaking, this is not "initializing" the array :)

Is it possible to skip this values and only assign the values 1, 2 and 3?

In C, Yes. Use designated initializer (added in C99 and not supported in C++).

int array[12] = {[0] = 1, [4] = 2, [8] = 3};  

Above initializer will initialize element 0, 4 and 8 of array array with values 1, 2 and 3 respectively. Rest elements will be initialized with 0. This will be equivalent to

 int array[12] = {1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0};   

The best part is that the order in which elements are listed doesn't matter. One can also write like

 int array[12] = {[8] = 3, [0] = 1, [4] = 2}; 

But note that the expression inside [ ] shall be an integer constant expression.

An alternative way to do it would be to give default value by memset for all elements in the array, and then assign the specific elements:

int array[12];
memset(array, 0, sizeof(int) * 12); //if the default value is 0, this may not be needed
array[0] = 1; array[4] = 2; array[8] = 3;

Standard C17

The standard (C17, N2176) has an interesting example in § 6.7.9(37):

EXAMPLE 13 Space can be “allocated” from both ends of an array by using a single designator:

int a[MAX] = {
    1, 3, 5, 7, 9, [MAX-5] = 8, 6, 4, 2, 0
};

In the above, if MAX is greater than ten, there will be some zero-valued elements in the middle; if it is less than ten, some of the values provided by the first five initializers will be overridden by the second five.

Example

#include <stdio.h>

#define MAX 12

int main(void)
{
    // n2176, § 6.7.9(37)
    int a[MAX] = {
        1, 3, 5, 7, 9, [MAX-5] = 8, 6, 4, 2, 0
    };

    for (size_t i = 0; i < MAX; i++) {
        printf("%d\n", a[i]);
    }

    return 0;
}

Output:

1
3
5
7
9
0  <-- middle element, defaults to zero
0  <-- middle element, defaults to zero
8
6
4
2
0