Working on a fast piece (well, at least for beginner) now. One thing I noticed is multiple run-throughs at fast tempo can get me tired very quickly. I remember a suggestion (by YT channel The Independent Pianist) to alternate between faster tempo and slow tempo during practice. The faster run is done at a tempo that just slightly push oneself, and the slow run is done to re-check everything (rhythm, articulation, dynamics, pedal) is still done correctly.
Your Strategies for Bringing A Piece Within Your Skill Level to Tempo
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Animisha Practising the piece at a tempo that was too fast for me.
Hands up all those who've never done that!
What works for me is first playing only a short passage at a fast tempo, then stopping to review it. If all was well: tempo kept steady, not tensing up, phrasing, dynamics, the lot, than I may try it again just a tiny bit faster. When I push the tempo, the thing I look out for is extra tension. If it feels just as comfortable when I play it faster as it felt before, I know I'm on the right track. If not I need to slow it down again to find out what's going wrong.
When I start stringing the short passages together, I always need to be on the lookout for buildup of unnecessary tension.
I need to be very familiar with the piece so that it's almost second nature, and then I can slowly speed it up.
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- Playing small groups fast, then chaining them
- Trying a dozen different technical adjustments (looking ideas up online as well) until you find something that works at tempo
- Spending an hour of lesson time trying out the above with a teacher
- Video recording and mirrors to observe yourself to notice points of tension
- Telling yourself not to play with tension, playing piano/pianissimo, playing short bursts so you don't tense up with time, allowing yourself mistakes so you don't tense up anticipating mistakes
- Clicking up with a metronome, ideally once you know your technique should work at tempo
- Trying to play faster than the intended tempo: this can suss out inefficiencies and it can become "easier" at tempo if done right.
- Playing the offending passage hands separate
Pallas I'm not an accomplished pianist, but I'll offer up what I think. OK?
I think our technical threshold is what gives us a "speed limit", so to speak. By that, I mean, we can't play any faster than we've developed, technically speaking. If we play scales, accurately, at 70bpm, but start to hit wrong notes at 80bpm, that's going to be our speed limit until we develop our finger skills, technique. But, since scales are not as complex as some melodies, our scale "speed limit" may be optimistic for a more difficult piece. Granted, it is possible, even common, perhaps, for a player to learn a song, or piece, well enough to play it fast. But that may not translate to the next song.
Personally, I can play some blues based things on the piano faster than I can play scales, only because I've become familiar with those licks. Generally, I play ballads, maybe a few swing-tempo tunes, so speed isn't much of an issue for me. If I wanted to learn to play fast, I'd find a "live, sitting right next to me type" teacher who would tell me how to fine tune my fingering technique.
Followup question: if the overall tempo of a piece is limited by the fastest (few) measures, should those measures be worked on first and most when raising speed?
You haven't been playing for very long. From my own beginnings, I've found that I inevitably hit a speed barrier. I can manage a piece quite well at a slower pace, but as I gradually increase the tempo, I reach a point where I cannot go any faster. Trying to push beyond this limit, I start making mistakes and lose a lot of the musicality. When this happens, I have to significantly reduce the tempo to correctly play the piece again.
What I've learned to do in these situations is to accept my final tempo, at least for the time being, and move on to the next piece.
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... feeling like the pianist on the Titanic ...
I simply think that playing a piece correctly, musically, and quickly takes quite a lot of time and practice. And of those three, the first two are the most important to focus on as a beginner. A high tempo requires lots of movements to be automated, and automation happens by itself, if you practise dilligently. Which you do!
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... feeling like the pianist on the Titanic ...
iternabe alternate between faster tempo and slow tempo during practice
This is exactly how I do it - I don't follow all these Youtube channels because I just do as I think I should do, but that's what I do naturally 
I never practice it faster than my brain or fingers can handle though, because this is what happens in my case:
iternabe Working on a fast piece (well, at least for beginner) now. One thing I noticed is multiple run-throughs at fast tempo can get me tired very quickly. I remember a suggestion (by YT channel The Independent Pianist) to alternate between faster tempo and slow tempo during practice. The faster run is done at a tempo that just slightly push oneself, and the slow run is done to re-check everything (rhythm, articulation, dynamics, pedal) is still done correctly.
I've kept that video on my YouTube play list for fast tempo advice (the title has to do with metronome). For a long time, his method was my "go-to" method for increasing tempo, although towards the upper end of his chart the tempo would increase too much for me and I'd go back to 5 bpm at a time.
My new favorite method for increasing tempo is Dr. Molly Gebrian's interleaved clicking up.
I thought I wouldn't reply to this thread because I feel like many people are too obsessed with speed when it's usually not what they should be focusing on, but when I saw this I had to react:
Pallas In the thread where this post originated, the advice was to prioritize speed over accuracy, and I was wanting other opinions, because I disagreed.
You should never sacrifice accuracy for the sake of speed. Speed comes from control. Greater control means greater speed. You do not increase your max speed by practicing speed but by being able to fully control your movements and gradually compressing the time it takes to execute them. Your max speed gets better very gradually over time and there isn't any point in trying to force it.
Here is an anecdote which I told several times on the othe forum about my scale speed. I used to practice scales every day for several years (I stopped now because I'm prioritizing other things). Over those years I almost never practiced for speed. When I did speed runs it was only to check myself to see how fast I could go. All of my normal scale practice was slow and in total control. Guess what happened? Each year my max speed went up. This continued to this day and what once felt like "there is no way my fingers can ever move that fast" now feels like "this is so slow I'm falling asleep". Every "impossible" wall I faced eventually vanished without me ever having to force a higher speed than I was capable of.
It's fine to use the metronome to click up to the quickest tempo you can reasonably achieve but don't fret about it if you hit a wall. Practice your scales and arpeggios diligently and you will get faster over time. I promise. 
BartK don't fret about it if you hit a wall
This is great advice!
@Pallas it'll all happen in time (haha, no pun intended) as you improve your skills. I am truly amazed at how much I've improved as a pianist as an adult, over the past decade. In my head, my skill level as a teenager is where I plateaued but I am a much better pianist now. Just keep at it, and with time it'll naturally happen!!
BartK I have had some success "forcing" (cajoling?) myself to play faster. My max speed does go up gradually overall, but I quickly managed to play quite quickly and usually don't hit speed walls.
The way I do it is to think about the movement, how I'm feeling physically, and try to "choreograph" the movement at a slow tempo. Then, play very small groups of 1-4 notes or half a measure, say, at the intended tempo. And then to analyze where exactly you can speed things up, where tension is holding you down, or where you are using too much movement, or misalignment. I guess the troubleshooting process pretty much just gets better with time. But I find it to be more efficient than just practicing and hoping to get better.
Also, when you try to push speed - Try to keep it to the minimum amount of information possible. Take out everything that comes before and after the spot that is difficult, take out the other hand if it is easy. You can even hone in on a specific group of say 2-4 notes. Isolate the difficulty very precisely. For instance, while playing scales, you can only practice the thumb crossings for a while. I find this sort of practice to be very good at quickly increasing speed.
Pallas people have offered various strategies BUT sometimes the issue is just that you're not ready to play at that tempo yet. But since you have a teacher, hopefully you don't experience the latter issue!
I think it's very common for people to have a dream piece that's beyond their skill level, and of course they want to play it at the tempo of the recording they love, but it's way too difficult! In those cases, maybe you have to sacrifice accuracy to have the feel of the piece but you have a lot more fun playing it like that. It's not "good practice" but if you're an amateur just playing for your own enjoyment, it wouldn't be so terrible, there are lots of "bad practice" habits out there. I don't know if this is even related to your original question here or elsewhere 
Pallas changed the title to Your Strategies for Bringing A Piece Within Your Skill Level to Tempo .
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BartK I feel like many people are too obsessed with speed
Not me In fact I have been overly relaxed with speed... almost too relaxed to be honest. But it so happens that I decided that my current piece, which is supposed to be a faster piece, I should be able to play, well, a little faster, lol. So I thought I'd share the way I have been going about it... especially since coincidentally the discussion was happening in this forum!
My strategy is to play it slightly faster each day (but still well within my ability) while continuing to play it slower every now and then to ensure I still feel calm and serene with it and enjoy the process.
What I find fascinating is that whenever I play it at a slower speed than I am able to, I can actually correct mistakes BEFORE they happen. My finger moves towards the wrong key, my brain notices it, and corrects it STILL ON TIME so it doesn't break the rhythm. Win win!