Thank you for the replies.
Give space to the other person to blow off some steam, and s/he might be prepared for a discussion later.
How long has this been going on?
And how much exactly are you willing to compromise on that issue for the sake of a closure?
I didn't go into detail and maybe I should have been more clear. There are multiple conflicts that I feel have remained unresolved, and some that maybe have been addressed but not to the full extent to which I feel I need to be able to address them. So it's more of a dynamic/trend than an ongoing conflict, although the latter is also true.
What am I willing to compromise? I suppose I'd have to have a particular situation in mind to be able to go into detail. I guess more than anything I'd like the discussion to offer both of us clarity and resolution, for us both to understand where the other is coming from and to feel heard. I don't need to "be right" or gain any sort of artificial one-upness from it... I just want greater understanding and peace. The presence of repetitive conflicts suggests something's amiss and it would be nice to address it and learn from it.
You can't resolve it if you can't discuss it.
You have my sympathy. It's maddening. If their intention is to blow off steam then they should say so otherwise it's just annoying and likely to breed distrust between you. However, ignoring is a highly destructive strategy used by many. Good luck getting them to see the light.
Thank you. This person is someone who is very tenacious in their stance when it comes to feelings/beliefs but generally seeks to express understanding toward others. I think it's a clash of needs/personalities more than anything and a loaded/turbulent history that is still being re-written in the present, so to speak. But nonetheless, it feels quite awful.
How do you resolve conflicts and/or gain a sense of closure when the other party refuses to discuss it?
It's tough. I think if someone has the feeling that conflict is not a good thing or if they already feel that there is a conflict of interest, no chance of compromise, etc. then it's harder for them to want to talk about it. If they feel their voice is not being heard or their feelings are not being recognized or acknowledged, then they're likely to avoid conflict. Or if they feel or perceive they will not get their way, then they may think there's no use in talking about it. So, with these reasons, it's going to be difficult for them to want to deal with it. I think if she feels that opinion or feelings, even if they are wrong, are appreciated and respected, then she'll be willing to talk. But if she feels her views or feelings are not going to be seen as valid, then she will avoid it.
Very, very true. Thank you.
You can talk it out with another person who will listen without judgment.
You can write it out in a journal.
You can meditate on your need(s) that underlie the conflict.
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Siamese cat had a good question. How long has this been going on and what/if you are willing to compromise.
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Rite gave excellent reasons for why someone may not want to discuss a resolution.
Care to give more details?
I'm not sure what else to add. I think a part of it is a difference in needs. It may be that I simply need to talk about these things more to the level at which they feel uncomfortable. Compromise is essential, but the outcome can be subpar because whichever approach is taken, either they feel uncomfortable and pressed or I feel like things have been unaddressed. I might need to consider fulfilling that need in other ways. There's also the issue that I feel I can't bring things up, whether they're bothering me or of a more benign nature, which is related but also something else altogether. Ugh. I think in part it comes down to an incompatibility and there's only so much that can be done with that, and a rocky history.
Are they refusing to discuss it, or taking time to cool off and collect their thoughts?
I tend to do that.. take a while to settle down if I'm extremely offended or upset--and it's so as not to make things worse.
Thank you for the perspective.

It's a little bit of both. Sometimes it's been a matter of "Can we talk about this later", except the later doesn't always come about and I hate to feel as though I'm nagging (since I'm more likely to be the one asking to talk about it in the first place).
I don't know guys (gals). I don't mean to sound as though they're being stubborn or resistant, or if they are that those reactions aren't in response to something I'm doing or saying. I'm trying to do what I can but I have my shortcomings in this area and I can understand if the resistance/reluctance is in part due to that in addition to what I've said about differing needs. I guess I'm at the point where I welcome doing what I can to improve it but I also realize the situation may not change much and I'm looking for a way to find peace anyway. I can't stand the strain and tension I feel from it. It drains me and it's wearying.