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Consider two definitions of a cubic region,

a = 5;
region1 = ImplicitRegion[0 <= x <= a && 0 <= y <= a && 0 <= z <= a, {x, y, z}];
region2 = Cuboid[{0, 0, 0}, {a, a, a}];
RegionPlot3D /@ {region1, region2}

enter image description here

We can test whether certain points fall inside the regions

{0, 0, 0} ∈ region1
{0, 0, 0} ∈ region2
(* True *)
(* True *)

But when I try to make a DiscretizeRegion object for use in an FEM NDSolve calculation, one is clearly better than the other:

{region1dis, region2dis} = DiscretizeRegion /@ {region1, region2}

enter image description here

This is clear visually as well as in region tests,

# ∈ region1dis & /@ {{0, 0, 0}, {.04, .04, .04}, {.4, .4, .4}}
# ∈ region2dis & /@ {{0, 0, 0}, {.04, .04, .04}, {.4, .4, .4}}
(* {False, False, True} *)
(* {True, True, True} *)

I can't find an option for DiscretizeRegion that fixes this. Not AccuracyGoal or PrecisionGoal or MaxCellMeasure, or any of the Methods listed in this post, this post, or this post. If this is a duplicate of one of those, my apologies

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4
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This is a tangential answer to your question, but perhaps it may be of interest nonetheless, in the spirit of providing an alternative approach.

In the case of a cubic region you may be interested in using the bounding box option to DiscretizeRegion, rather than trying to coax it to include all the edges (which I was not yet able to do either).

For instance:

region3dis = DiscretizeRegion[
   FullRegion[3], {{0, a}, {0, a}, {0, a}},
   MaxCellMeasure -> 20
 ]

Mathematica graphics

# ∈ region3dis & /@ {{0, 0, 0}, {.04, .04, .04}, {.4, .4, .4}}
(* Out: {True, True, True} *)

Incidentally, this works well with the FEM element mesh generators as well:

Needs["NDSolve`FEM`"]
ToElementMesh[FullRegion[3], {{0, 5}, {0, 5}, {0, 5}}, MaxCellMeasure -> 20]

(* Out: ElementMesh[{{0., 5.}, {0., 5.}, {0., 5.}}, {HexahedronElement["<" 8 ">"]}] *)

%["Wireframe"]

Mathematica graphics

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