While the question you ask is theoretically simple, there are a lot of factors that need to be taken into account when dealing with this. The difference between the two builds isn't just "Oh, I'm going to change out a couple of levels and have basically the same character!" There's a lot of differences, and there can be a lot of things that change between the characters. For simplicity, all of the characters will share the same race and ability scores. This means that I can calculate the same attribute bonuses how I like, and basically make the character have the skills we need.
The first decision is to assume we are going a strength-based character. We could do a bow Fighter, and it would work well especially with the long-range of the bow, but I am still trying to max out the Champion aspect of the Fighter/Wizard, and using a Str-Based character with a Greatsword means that the critrange of 19-20 is helped out that much more by 2d6 melee over 1d8 ranged damage from a Longbow.
The goal for the Knight is to get to basically max Str/Int for damage purposes, and then either 18 Con + Tough for max HP, or 20 Con for 25 less HP than the tough mode and gain +1 Con Mod.
For the Fighter/Wizard, we want to maximize damage. While this class should still have a fair bit of melee ability, we need to maximize the offensive abilities best we can. Trying to max out Str/Int first is good, and then having the rest of the ability score increases (which there are only 5 of) to round out the HP gives the Fighter/Wizard a significantly worse HP.
We are going to use a Mountain Dwarf. Since we only need Int, Str, and Con as our stats to max now, since Wis, Cha, and Dex don't matter, we want to maximize those values. Mountain Dwarf gives +2 to Str and +2 to Con, so we can start out with a 17/8/17/15/8/8 character. With the Eldritch Knight, it's impossibly easy to get 11 points of Attributes out of the 7 attribute increases.
Itemization will be the same. They will both have 2 +1 Longswords/Shortswords and +1 Full Plate (for defensive purposes). This will give them max AC and max damage for now.
The target will be an Ancient Red Dragon. It has 22 AC, which to my knowledge is the hardest thing to hit at the moment.
From there, it becomes a simple measure of planning and building the characters.
Elder Rich
Level 20 Mountain Dwarf Fighter (Eldrith Knight)
- Max HP: 244 (10 + 19*6 + 20 * 4 + 20 * 2)
- Proficiency: +6
- Strength: 20 (+5)
- Dexterity: 8 (-1)
- Constitution: 18 (+4)
- Intelligence: 20 (+5)
- Wisdom: 8 (-1)
Charisma: 8 (-1)
Start: 17/8/17/15/8/8
- 4th: Dual Wielding
- 6th: Heavy Armor Master (Str to 18 (18/8/17/15/8/8))
- 8th: Tough
- 12th: Str to 20 (20/8/17/15/8/8)
- 14th: Con to 18, Int to 16 (20/8/18/16/8/8)
- 16th: Int to 18 (20/8/18/18/8/8)
- 19th: Int to 20 (20/8/18/20/8/8)
Features:
Two Weapon Fighting (Fighting Style), Second Wind, Action Surge (Two Uses), Weapon Bond, Extra Attack (3), Indomitable (three uses), Weapon Bond (Eldritch Knight), Improved War Magic (Eldritch Knight), Eldritch Strike (Eldritch Knight), Arcane Charge (Eldritch Knight)
- Spell Slots: 4/3/3/1
Feats: Tough, Dual Wielding, Heavy Armor Master
Main Hand: +1 Longsword (+12 Attack, 1d8 + 12 Damage (16.5 Avg. Dmg.), 2d8 + 12 Crit (21 Avg. Dmg. Crit))
- Off Hand: +1 Longsword (+12 Attack, 1d8 + 6 Damage (10.5 Avg. Dmg.), 2d8 + 6 Crit (15 Avg. Dmg. Crit))
AC: 20 (+1 Full Plate, Dual Wielding)
Rich is probably going to be able to deal a fair bit of damage. Each single Main Hand Longsword strike will deal an average of 16.5 Damage on a hit. Against a Red Dragon, that's a 55% chance to hit (and 5% to crit), so 9.3 damage per swing accounting for accuracy (and crit). The offhand is 10.5 damage on average, so a 6 damage per swing accounting for accuracy (and crit). That means every attack action is going to be 15.3 damage per swing. Considering that Rich will get 4 attacks per turn, this is 61.2 DPR on just a basic set of attacks. With an Action Surge, that damage jumps to 122.4, which is approximately 22% of an Ancient Red Dragon's max HP. Two turns right after each other, Rich Alone could easily bloody the Dragon with slightly lucky rolls.
The main hand weapon used to be a +1 Greatsword. Compared to the current build, it did less damage on Rich's turn (10.8 damage per Attack action, 43.2 damage per turn vs. 15.3 per Attack Action, 61.2 per turn), but had a higher reaction -> opportunity attack damage. This is because it had an average damage of 19 Damage (Crit 26). These extra points of damage could stack up well, but to be honest, the lack of 2.5-5 points of damage on the 1 reaction per round loses out to the heavily increase DPR thanks to the dual wielding. The Greatsword also had a higher Improved War Magic damage, thanks to being able to attack only with your main-hand weapon. Again, while the loss of 2-5 damage per Spell -> Attack is known, the heavy increase in overall DPR from the multiattack functionality of Fighters is much more useful.
Fig Tier Wiz
Level 20 Mountain Dwarf Fighter (10)/Wizard (10)
- Max HP: 184 (10 + 9*6 + 10*4 + 20*4)
- Proficiency: +6
- Strength: 20 (+5)
- Dexterity: 10 (+1)
- Constitution: 18 (+4)
- Intelligence: 20 (+5)
- Wisdom: 8 (-1)
Charisma: 8 (-1)
Start: 17/8/17/15/8/8
- 4th: Heavy Armor Master (Str to 18 (18/8/17/15/8/8))
- 6th: Str to 20 (20/8/17/15/8/8)
- 8th: Int to 17 (20/8/17/17/8/8)
- 14th: Int to 19 (20/8/17/19/8/8)
- 18th: Con to 18, Int to 20 (20/8/18/20/8/8)
Features:
, Second Wind, Action Surge (One Use), Improved Critical, Extra Attack, Remarkable Athlete, Indomitable (one use), Great Weapon Fighting (Additional Fighting Style); Arcane Recovery, Arcane Tradition, X Savant, 3 Arcane Tradition Features (Arcane Ward, Projected War, Improved Abjuration || Sculpt Spell, Potent Cantrip, Empwered Evocation)
- Spell Slots: 4/3/3/3/2
- Feats: Heavy Armor Master
OPTION 1:
- Main Hand: +1 Greatsword (+12 Attack, 2d6 + 12 Damage (19 Avg. Dmg.), 4d6 + 12 Crit (26 Avg. Dmg. Crit))
- Feature: Defense (Fighting Style)
- AC: 20 (+1 Full Plate, Defense)
OR
OPTION 2:
- Main Hand: +1 Shortsword (+12 Attack, 1d6 + 12 Damage (15.5 Avg. Dmg), 2d6 + 12 Crit (19 Avg. Dmg. Crit))
- Off Hand: +1 Shortsword (+12 Attack, 1d6 + 6 Damage (9.5 Avg. Dmg), 2d6 + 6 Crit (13 Avg. Dmg. Crit))
- Feature: Two Weapon Fighting (Fighting Style)
- AC: 19 (+1 Full Plate)
So here I've given out 2 builds for Fig that both can work. We only have 5 Ability Score increases, so it is much more difficult to max out Str and Int and get a high Con score. Technically, Heavy Armor Master can be exchanged for any feat that gives +1 Str, Con, or Int, but the DR 3 on non-magic weapons seems like a huge deal.
Option 1 is what I had originally come up with for Fig and Rich. The idea is to max out damage thanks to Str and Int, and use a Greatsword for combat. Because of that, you can take the Defense Fighting Style for +1 AC. The damage for this character is 10.8 Damage per attack accounting for accuracy against an Ancient red Dragon, and therefore a 43.2 Damage per turn.
Option 2 is more like Rich, wanting to use two weapons. The best/only options for max 1-handed light damage is Shortsword basically. As with Rich, the damage is greater over the course of a round than using the Greatsword because of the multiple hits you get in per attack, but you have a lower amount of damage on a reaction, by about 2 points (3.5 points on a hit, 7 points on a crit). This build sacrifices 1 AC to get an increase of damage at close range.
(As an aside, Option 2 could also possible use an 18/8/18/20/8/8 final ability score with 2 +1 Longswords and Dual Wielding. I checked the numbers, and while you do get +1 AC thanks to dual weilding, you actually lose damage overall. This is because while you do get to go from 1d6 weapons to 1d8, you lose +1 to your Attack Modifier and +1 to your damage. The mainhand would be +11 to attack, 1d8 + 11 Damage (15.5 Avg. Dmg), 2d8 + 11 Crit (20 Avg. Dmg. Crit)). As you can see, the main hand would basically lose 5% on normal attacks, while the Average Crit damage increases by 1. This is the same for the off hand. The only difference is basically that you lose 5% to-hit to gain a marginal benefit when you crit.)
So right off the bat we see that Fig is lacking in a couple of areas compared to Rich. He's already at a much lower HP thanks to the levels in Wizard only giving +4 to HP per level instead of +6. In addition, the lack of ability score increases means that hitting 18 Con + Tough or 20 Con is almost impossible without sacrificing damage. With 60 HP less than Rich, it's a fair bit of damage that is not turned lethal.
Second, the lack of Action Surges and Extra Attacks means that Fig won't be able to attack more than twice a turn. This is huge when you compare the two characters. The lack of Dual Weilding for Fig means no "natural" dual wield longsword, and therefore a fair bit of loss of damage. On top of that, you can only deal ~28.2 damage per turn, ~56.4 with your single Action Surge. The damage that Fig is going to be putting out is on the level of a 10th level fighter, not a 20th Level Fighter/Wizard.
Speaking of the Wizard side, you are still not a super-powerful spellcaster when it comes to spells. What you give up from the Wizard Side to "splash" 10 levels in Figher means that you only have access to 5th level spells. While not an issue for the most part, 5th levels spells will need to greatly make up for the damage deficit that the lack of extra attacks the fighter should have incurs. You also "only" have 2 5th level slots and 2 4th level slots over the Knight. I put "only" in quotes because while I can attempt to downplay those slots, they represent a level of power the Knight cannot achieve. If we are talking about pure damage, the Knight can cast 1 4th level spell, but will probably want to prefer to use his spells as support spells to further lock-down his targets while he beats the heck out of them. The Wizard can use those 4 extra slots to provide a lot of utility to the party.
For comparison of the spell casting abilities, we need to look at the damage between the classes. Because spell damage is only based onthe spell, not your For now, I think the best option to check for spell damage is going to be the Spell Damage Comparison Chart put out by Rhaegar83 on /r/dndnext.
Looking at the spells, it's clear that no 5th level spell can match the single-turn damage of a 20th Level Eldritch Knight attacking the crap out of a single target. While you do get access to some nice 5th level AoE spells like Cone of Cold and Flame Strike, the Eldritch Knight can easily hit 2 targets for similar Damage if he is able to get between the two using his move.
The end result is that Rich, the Eldritch Knight is going to end up tankier (HP and AC) and able to deal more single-target damage thanks to his quad-attack. Fig is going to be able to deal more AoE damage, single by virtue of having access to 5th level spells, as well as 2 more 4th level spells slots. Fig is also going to be able to be more of a controlling presence on the battlefield, if he is going to take at least some support/control-y spells.
I agree that the part about recharging is a bit ambiguous, because it's part of a two-sentence paragraph:
While the ward has 0 hit points, it can't absorb damage, but its magic remains.
Whenever you cast an abjuration spell of 1st level or higher, the ward regains
a number of hit points equal to twice the level of the spell.
If these were written in the other order, I don't think there'd be any potential for confusion, but this way, I can see someone reading the second sentence as conditional on the first. But, I don't think it's meant to be: if that were the case, the second sentence should be more clear about that.
So, I'd read it as your two spells (4th level and 5th level) recharging the ward to 18hp. And, this is also kind of sloppily worded, but presumably the ward's initial hit points are also the maximum — you can't "recharge" beyond that.
This reading is confirmed as the design intent in a tweet by co-designer Jeremy Crawford — thanks to wax_eagle for checking.
Note that the ward does not mention temporary hit points. It is a stand-alone rule, and it should be read just on what it says. Even if the ward acts similarly to having temporary HP in some ways, it doesn't say that they are, so they aren't — and can work differently in some of the details. (And, crucially, you might have both the ward and temporary hit points.)
For what it's worth, this does seem pretty awesome: a 10th level barbarian with a 20 con has 125 hit points. A level 10 abjurer with a 16 con has 72, and a 25 hp buffer — and if she uses all of her spell slots casting abjuration spells, that can recharge up to 82, for an effective total of 179. If one of those spells is Stoneskin, she has the same resistance to normal weapon damage as the ranging barbarian, although probably not as high an AC without making a lot of the castings Shield.
On the other hand, a 10th-level Moon Druid (say, around 83 hp) can shift into Ankylosaurus form twice between short rests, giving what's effectively two 68hp buffers and also you are a dinosaur. I expect once we get into Deep 5e Optimization Territory, there will be plenty of other things which outstrip the arcane ward.
Best Answer
Paladin 2 / Wizard (abjurer) X
You said you are willing to take at most a 2 level dip in another class for proficiencies. Have you considered taking these levels in Paladin? Assuming you want to keep the "I'm a Wizard!" flavour, taking your first 2 levels as a Paladin of the god of magic (assuming there is one in your campaign, of course) isn't really out of character. So here's my take on this build. I'll mostly cover melee options, since this is what the question is about, but you should still be able to do anything that another wizard could do, just 2 levels later. This gap can be great at certain levels (Fireball is a great damage upgrade at level 5), but it shouldn't be noticed too much on single target damage (Smite!). It's not the strongest build out there, but it's definitely good enough to be enjoyable!
Benefits of 2 paladin levels :
That being said, given equal gear, you should have the same AC as any other tank, since you're wearing the same stuff they are. As for your damage, the new Sword Coast Adventure Guide cantrips are there for you. Sure, they're not going to out-damage a fighter or a rogue, but they're still very good damage options. Since they require a melee weapon attack, they also trigger class features such as Divine Smite, so you don't miss out too much. Having two attacks could technically mean 2 smites per round, but the added damage of your cantrips will compensate for this quite decently.
Other thing to note is that the secondary damage from these spells automatically hits, no save and no attack roll required, which means that you can strike a minion to land free damage on the BBEG. These spells will thus give you decent damage round after round, leaving you with all your other spell slots to use as you see fit. Your 2 levels of paladin will give you the opportunity to convert your low-level spell slots into more damage as an option, and your wizard powers will allow you to turn these same spell slots into defensive abilities. Shield will grant you +5 AC for one round, which is awesome if you attract the BBEG's attention (especially given that your AC is already high). In Elemental Evil, there is also the Absorb Element spell, which gives you resistance against a single element, also for one round. This gives you defensive options for both physical and most magical attacks. As an interesting addition, both of these spells are abjuration-based, and will thus heal your ward for 2 HPs every time you cast them.
And after all this, you're still an almost full wizard. Your access to spells and features will be delayed for two levels because of multi-classing, which shouldn't reduce your melee effectiveness, but you will still have access to 9th level spells and have spell slots as a 19th level caster. While the Signature Spell capstone is good, the real treat is at wizard level 18 (character level 20 for you), so you don't miss out on much. Indeed, spell mastery will allow you to have 2 low-level at-will spells, which gives you infinite Shield spells, for instance. This should allow you to use your slots for more offensive options, since your defence is free. Infinite Shield spells also means that your arcane ward is always topped off after every combat if you take a minute or so to recharge it after each fight.
Your melee power also scale quite decently if we compare them to a tank fighter (longsword + shield, defensive fighting style) of equivalent level :
Of course, a fighter has abilities to increase his damage, but so do you (thank you, Divine Smite)! Since the fighter has more attacks, he's more likely to deal some damage each round, whereas your damage all hinges on your single attack landing. This will be especially important if the fighter can get his hands on a weapon that deals extra damage (like a flame tongue), an ability that would be mostly wasted on your single attack per round. The fighter also only requires only one good stat to be 100% effective, while you will need both strength and intelligence. Since your intention is to go melee, I'd consider maxing strength first, especially considered you'll use your spells mostly defensively or in a utilitarian fashion (so lower save DCs are not as bad).
Feats to consider :
Other things to consider :