What is the fallacy of ignoring the other person's refutation and asserting victory?

Even if you present your sibling with facts he agrees with, you will lose the argument. In my experience, people never change their beliefs by being punched in the head with facts. Most people believe what they believe because they want to believe what they currently believe. Facts are not important. Michael Shermer made this addition to Cognitive Dissonance Theory in his book, “Why People Believe Weird Things.” So, if your goal is to change another person’s belief, I think you must use a different approach.

Peter Boghossian suggested a strategy to change a person’s belief. To be successful, he said the person whose belief you want to change must reconsider how he arrived at the belief under discussion. If your goal is to change his mind, as distinct from pontificating (which is better done in front of a mirror), then you need to get him thinking about how he arrived at the belief.  

Boghossian’s book, “How to Have Impossible Conversations,” is an excellent manual on how to do this. In my experience, people never change their beliefs by being punched in the head with facts. Most people believe what they believe because they want to believe what they currently believe. Facts are not important. Michael Shermer made this addition to Cognitive Dissonance Theory in his book, “Why People Believe Weird Things.” So, if your goal is to change another person’s belief, I think you must use a different approach.

Peter Boghossian suggested a strategy to change a person’s belief. To be successful, he said the person whose belief you want to change must reconsider how he arrived at the belief under discussion. If your goal is to change his mind, as distinct from pontificating (which is better done in front of a mirror), then you need to get him thinking about how he arrived at the belief.  

Boghossian’s book, “How to Have Impossible Conversations,” is an excellent manual on how to do this. He suggests asking questions. For example:

“I’ve come to a different conclusion and I’m having a hard time understanding where you’re coming from. I assume you must know some things about this subject that I don’t. Could you tell me more about where you’re coming from on that so I can understand better?”

The more ignorance you admit, the more readily your partner in the conversation will step in with an explanation to help you understand. And the more they attempt to explain, the more likely they are to realize the limits of their knowledge and epistemological errors made along the way.

If you ask someone a direct question and he obfuscates or refuses to answer, ask him to ask you the same question, and you answer it. Other Boghossian suggestions:

“That’s an interesting perspective. What leads you to conclude that?”

 Say, “I’m skeptical,” not “I disagree.”

 “On a scale from 1 to 10, with 1 being no confidence and 10 being absolute confidence, how confident are you that belief is true?”

“I’m not sure how I’d get to where you are, at a X. I want to see what I’m missing. Would you help walk me through it?”

“I am not trying to convince you of anything. I’m curious and would like to ask some questions to learn more.”

The idea is instead of people holding a belief because they think they should hold that belief, reverse it and claim to hold your belief and wish you could stop believing—if only the discussion partner could show you the error of your ways. The point is, you want to get them thinking about the process that led to the conclusion and not about the conclusion itself. 

All of this deals with the Fallacy of Subjectivism. Subjectivism is not only a way of adopting conclusions on subjective grounds, but also — and probably more often — a way of evading the grounds. Some people have perfected the skill of ignoring what they don’t want to see, and most of us indulge in this habit from time to time. Heuristics are hell. If I put the statement into a proposition, it takes the form: “I don’t want to accept p. Therefore p isn’t true.” That’s the fallacy of subjectivism. Another in use by your sibling is the Fallacy of Diversion. And there are more. As "they" say, so many fallacies, so little time.

Ignoring refutations of a point and just continuing on would be avoiding the issue . I think you may be able to argue that baseless assertions of victory in an argument may also fall under that, as it's being used to further avoid the refutations and shift the focus to something else.