How do I handle my neighbor's dog peeing on my lawn?

There are several brown ugly splotches on my lawn and on a few occasions poop left without being picked up. This morning I just so happened to catch an older woman walking her dog around my area, and when I backed out my car she promptly disappeared. I found her around the corner at her house and said, “I don’t know if it’s your dog, but do you think you could not have him pee on my lawn?” I was polite. She said, “This is ludicrous!” and said I should have a fence if it bothered me and I was being unreasonable because I can’t control where her dog does his business. I told her, “I understand there are other dogs, I’m not accusing you as the only one,” but she threw up her hands and walked away.

That escalated quickly. I honestly don’t care about my lawn THAT much to fight about it, but it’d be nice to not have to worry about poop and pee on it, ya know?

Any advice?