Looks fungal to me. Probably anthracnose, also called black spot, as has already been suggested. The disease is most prevalent in the spring when the conditions are cool and moist, but can also happen in the summer when conditions are favorable (prolonged rainy periods like you're having now). This fungus thrives in damp conditions.
Here are some tips.
Prevention
- Grow your tree in a coarse-grained, well-drained soil. Avoid lots of fine-grained material (1/8" or less) that will keep your tree overly damp in a bonsai pot.
- When you water, water in the morning and don't water the foliage. Check again in the late afternoon, but avoid watering at night.
- Grow your elm in the full sun or light (<25%) shade.
- Adopt a dormant spray regiment to avoid fungal problems in the spring.
- Make sure that you have space between your trees and that they're not too crowded together.
Solutions
- Prune back infected branches to a healthy section and dispose of in sealed plastic bags. Make sure to cut back beyond ALL the diseased leaves. Even those with just one spot.
- Disinfect your pruning tools to keep the disease from spreading to healthy plants. Wipe the blades off with alcohol or a household disinfectant and allow the blades to dry before using.
- Clean up fallen leaf debris and dispose of in plastic sealed bags.
- Spray with Daconil. Use a good pump sprayer that delivers a fine mist and get all the leaves. Spray when the foliage is dry.
As long as the fungus is restricted to the foliage, this is easily treatable. If it spreads to the branches and trunk it will result in wholesale collapse of entire branches. At that point it's really not curable so it's important to take timely steps. Your case looks totally treatable.
Scott