With the technology available to them at the time? No. At least not on the Western Front. It is important to realize that there is a vast qualitative and quantitative difference between the Western Front and the Eastern Front.

When people think of trench warfare, they usually think of the Western Front, so that's what I am going to stick to. The short answer is no. The long answer is more complicated, and involves doctrinal issues and technology.

With the vast increase in the power of the defensive over the offensive that we saw in the years immediately preceeding World War One, and the relative lack of developments in offensive capabilities, significants breakthroughs were more or less impossible when faced with defensive belts dozens of kilometers deep. Reinforcements could be brought up quicker than you could hammer your way through defences due to the lack of mobility on part of the offensive force. While the artillery did mitigate this problem somewhat, it was a double-edged sword. You could be bombarded the same as you bombarded your enemy.

As another poster also correctly points out, there were numerous attempts at solving the gordian knot of the Western Front. In fact, we do see a progression of increasingly complex battlefield tactics employed from the wildly uncontrolled battles of the frontiers to the highly complex German Spring Offensives, and although it may look, from a laymans perspective, that all they did was simply add more men to try and push through, this is not the case. Significant developments in artillery occured, as did infantry tactics. Instead of using human wave, suicidally storming into massed machine gun fire, infantry would try to identify strong-points and circumvent them, (or call down artillery to destroy them) in what was called infiltration tactics. Aircraft was also used with more sophistication. For instance, during Verdun, they tried to hamper the french ability to conduct reconnaissance in order to deny them any knowledge of the impending offensive.

All these developments ran into the eternal problem of the first world war; sustaining an offensive. One of the reasons the German Spring Offensive failed, was the inability for the offensives to be sustained. They ran out of steam, so to speak. Sure, they had penetrated enemy defenses in places, but they could not carry that penetration through and convert it into a general breakthrough.

Sources:
Georgii I. Isserson: Evolution of Operational Art
A. A. Svechin: Strategy
Peter Hart: The Great War
Robert Foley: German Strategy and the Road to Verdun

There are quite a few answers in our back catalogue on the tactics/strategies employed in the First World War, but this recent answer from u/bodie87 covers the rationales and limitations that led to this form of warfare on the Western Front.

We've had lots of great discussions on trench warefare warfare! There's always more to be said, but while you're waiting here are a few:

• On landing soldiers behind enemy lines by /u/thefourthmaninaboat

• On navel naval invasion also by /u/thefourthmaninaboat

• In fact, you might just want to check out a whole series of responses on amphibious warfare by /u/thefourthmaninaboat

• And last but not least, on mobility on the eastern vs western front by u/DuxBelisarius

• Edit to add one more! A recent thread on WWI tactics by u/bodie87

2nd ETA: I corrected my spelling errors/autocorrect mishaps. That's what I get for rushing a comment out!

I'd start with Norman Stone's The Eastern Front and the recent book The Russian Army in the Great War by David Stone. The Eastern Front, unlike the Western, really was a mobile war of maneuver. There are complicating factors of a sliding scale of logistics with Germany-Russia-AustriaHungary-Ottoman Empire in that order, which in turn sets the pattern of combat. Germany has the best army because of the best logistics, Russia steamrolls everything not-Germany for the same reason. Against the Germans the Russians can win on the defensive and won a single offensive battle at a tactical level, but the superior logistics factor made strategic offensive victories nearly impossible for them.

What makes the differences between the Eastern and Western Fronts starts with the logistics, then with the sheer size of terrain and the front. The Polish and Galician campaigns were fought by armies that were thin on the ground relative to the terrain they had to cover, but sharing the same juggernaut factor as the Western mobile campaigns where armies slaughtered more quickly than their commanders could adapt to the shifting fluid lighnting-fast decisions needed to capitalize on tactical and operational gains to achieve strategic results.

Holger Herwig's The First World War: Germany and Austria-Hungary likewise deals with these issues, and notes that the command and control issues that dogged the Allies at points in the West were every bit matched or exceeded by those of the Central Powers. Germany underwent a command collapse in the opening campaign of World War I in the West that the French and the British did not. In Joffre commanding his armies according to a plan and Moltke letting seven generals do as they pleased were the lines decided, but the result was Germany holding an overextended front where it could neatly defend and the enemy had no choice but to try to get more favorable terrain.

The Aufstragtaktik myth was one concocted more after the war than during it, and it served as an ad hoc justification and a pretense that nothing actually did go wrong. It was just that: a myth. In reality, Moltke the Younger proved utterly incapable of commanding armies of the size he did or serving as a general in even a nominal sense, and the German Army, to use one example, had 600,000 men in two armies in the path of 150,000 men in the BEF and in two battles the two corps of said BEF fought separately and proved far more efficient than the size and relative firepower of the German Army indicated should have been so on paper.

This was only partially the BEF's professional status, the other part was that Kluck and Bulow were too old for the job and their tactics were headlong attacks into prepared defensive positions in close order in perfect 1871 fashion with perfect 1871 casualties ensuing. And in this they were not atypical of the German Army in reality East or West. A more mobile war would have exposed far more of the failures of the German officer corps to adjust to the realities of the war it faced, which is why the rigid defense (aka trench warfare) served as their one genuine short term strategic 'decision' of some adeptness.

Hindenburg and Ludendorff adopting flexible as opposed to 'Not One Step Back' style defenses was what started the process of deeper changes. Tanks and combined-arms warfare of the WWII sort were in their infancy and marked qualitative tactical changes from what preceded them. Herwig, who stands unique as one of the very few individuals in English-language WWI writing to deal in detail with the realities versus the myth of the Kaiserreich and what it did and did not do, uses primary sources to illustrate bluntly that the 1918 offensives were more of a hollow success that marked the apex of tactical virtuosity but rigid operational planning and no strategic concept whatsoever, meaning that the mobile campaigns of 1918 lost Germany the war by killing the remaining high-quality troops it had for a bunch of salients it could not hold nor defend.

The 'infiltration tactics' were more of a measure of desperation, and the reliance on firepower was a result of the losses of 1914-7 sapping Germany's manpower and the hollow starvation state behind the lines meaning that desperation served as a substitute for skill with the exact success that would be expected from willpower vs. firepower.

Germany was not innately better at war-fighting in the World Wars scale than its enemies, it had very deep and profound military issues and it never really solved them under the Weimar Republic or the Nazis.

Sources:

The Eastern Front, Norman Stone.

The Russian Army in the Great War, 1914-7, David Stone

The First World War: Germany and Austria-Hungary 1914-18, Holger Herwig

The Marne, Holger Herwig

Yes, absolutely. I want to try broadening the strategic aperture here and my answer focuses on the role of economic warfare and the influence of the United States.

I think it's important to note that both sides had and implemented a plan for winning the war that did not focus on incremental gains in trench warfare. In fact, both sides hoped to win the war by strangling the other economically. For the allies, the goal was to do this through the naval blockade. For the Germans, the goal was to do this through the U-Boat campaign. Both of these were implemented early in the war and tightened as it went on. Both of these could plausibly have ended the war sooner.

Before getting to that, I want to take a step back and look at how the war actually ended. Take a look at the map of how territory changed hands. Unlike WWII, the Germans had not been pushed back very far at all at the time of the armistice. Germany gave up for a variety of reasons, but probably the most important were the shortages of food caused by the Allied blockade and the change in the alliance picture (US entered on the allied side, the other Central Powers were collapsing). Germany's situation moving forward was hopeless.

It didn't work out according to the German plan, but the Germans hoped to knock Britain out in much the same way by using U-Boat warfare to starve the British and destroy the UK's economy. The Allies would collapse, surrender, and Germany would win.

It's also important to note at the strategic level that the two sides had been sending out serious peace feelers for years. Before the US entered, Woodrow Wilson believed that he had a serious chance of negotiating an end to the war. In some sense, then, the answer to your question is that there were alternatives (blockade, U-Boats, knock out allies), these alternatives were tried, and they did end the war.

So, why couldn't all of that happen faster? Of course, economic warfare takes time but both sides could have pursued their policies much more aggressively at an earlier point. The answer as to why they did not hinges critically on the role of the United States. The United States was neutral, powerful, and plausibly capable of deciding the fate of the war. The U.S. was also opposed to both U-Boat warfare and the blockade at least as far as these infringed on American interests.

Edward Grey, the British Foreign Secretary, wrote of the strategic situation in his memoirs: "Blockade of Germany was essential to the victory of the Allies, but the ill will of the United States meant their certain defeat … It was better therefore to carry on the war without the blockade, if need be, than to incur a break with the United States about contraband and thereby deprive the Allies of the resources necessary to carry on the war at all or with any chance of success." In short, the British calculus was that they could only push the blockade so far without losing American support and thus the far. So the most potent weapon in their arsenal had to be used hesitantly and incompletely. Over time, and as pro-Allied sympathies in the US became more pronounced, Britain tightened the blockade but it had to do so haltingly and hesitantly.

The German position was similar. Germany was not hoping for American support in the way that the British were, but the Germans at least hoped to avoid the U.S. joining the other side. This meant that U-Boat warfare had to constantly be tempered to avoid antagonizing the United States. Just like the British, the Germans could not attempt a decisive blow without risking adverse American action. Over the early years of the war, Germany vacillated on the issue. Eventually, Germany decided to gamble on resuming unrestricted submarine warfare in early 1917 because they thought that they could knock Britain out of the war before U.S. involvement began to matter and they had few other good options. In retrospect, this was clearly the beginning of the end.

It's interesting to speculate on what would have happened if one or both sides had reached for the decisive economic weapons earlier. It's not hard to believe that it would have correspondingly shortened the war without the need for so much inconclusive fighting on the Western Front.

Sources:

-Nicholas Lambert, Planning Armageddon: British Economic Warfare and the First World War

-Karl Birnbaum, Peace Moves and U-Boat Warfare

-Edward Grey, Twenty-Five Years