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In my program, I need to check if a variable equals 1, 2 or 3 and depending on the result, perform a different method:

if (phase.equals("1")) 
 PhaseOne.performPhase(inputParser.getSource(), inputParser.getTarget());
 else if (phase.equals("2")) 
 PhaseTwo.performPhase(inputParser.getSource(), inputParser.getTarget());
 else 
 PhaseThree.performPhase(inputParser.getSource(), inputParser.getTarget());

This code is so simple and basic but I really don't like it. Of course I could use switch conditions but it would, in my humble opinion, just display the same basic function in a different way.

My question is: is there a way to implement the function in an elegant and expandable way?

FYI, I already red this post but I did not find an answer which fits to my question.

In my opinion, the accepted answer on your linked question fits you very well. Store references to the functions in the map:

Map<String,BiConsumer<T,U>> map = new HashMap<>();
map.put("1",PhaseOne::performPhase);
map.put("2",PhaseTwo::performPhase);
map.put("3",PhaseThree::performPhase);
map.get(phase).accept(inputParser.getSource(), inputParser.getTarget());

Replace T and U by the types of inputParser.getSource() and inputParser.getTarget().

With this approach, the Phase… classes don't need a common superclass or interface.

If your PhaseOne/PhaseTwo/PhaseThree classes all implement the same interface (let's say Phase), and the method performPhase is defined on the interface you could do the following:

final Phase targetPhase;
switch(phase) 
 case "1": targetPhase = myInstanceOfPhaseOne; break;
 case "2": targetPhase = myInstanceOfPhaseTwo; break;
 case "3": targetPhase = myInstanceOfPhaseThree; break;
 default: throw new IllegalStateException("Unrecognised phase "+phase);

targetPhase.performPhase(inputParser.getSource(), inputParser.getTarget()));

Another option is to create a Class for each Phase and an IPhase Interface for them to implement.
Create a List<IPhase> with all your different Phase instances.
Run a loop and if the id matches, execute the overwritten method.

public interface IPhase 
 public void performPhase();
 public String getId();


for (IPhase phase : phasesList)
 if (phase.equals(phase.getId()))
 phase.performPhase();
 // either break or continue the loop