Definition of a boundary value problem:

pde = {-(1050000/13) D[u[x, y, z], z, z] - 
    1050000/13 D[u[x, y, z], y, y] - 2625000/13 D[w[x, y, z], x, z] - 
    2625000/13 D[ v[x, y, z], x, y] - 
    3675000/13 D[u[x, y, z], x, x] == 
   0, -(1050000/13) D[v[x, y, z], z, z] - 
    2625000/13 D[w[x, y, z], y, z] - 3675000/13 D[v[x, y, z], y, y] - 
    2625000/13 D[u[x, y, z], x, y] - 1050000/13 D[v[x, y, z], x, x] ==
    0,
  -(3675000/13) D[ w[x, y, z], z, z] - 
    2625000/13 D[v[x, y, z], y, z] - 1050000/13 D[w[x, y, z], y, y] - 
    2625000/13 D[u[x, y, z], x, z] - 1050000/13 D[w[x, y, z], x, x] ==
    0}

bcs =
 (* Dirichtlet bcs *)
 {u[50, y, z] == 0,
  v[x, 0, z] == 0,
  w[x, y, 70] == 0,
  (* Neumann bcs*)
  (D[u[x, y, z], y] + D[v[x, y, z], x] == 0) /. {y -> 50},
  (3 D[w[x, y, z], z] + 7 D[v[x, y, z], y] + 3 D[u[x, y, z], x] == 
     0) /. {y -> 50},
  (D[v[x, y, z], z] + D[w[x, y, z], y] == 0) /. {y -> 50},
  (3 D[w[x, y, z], z] + 3 D[v[x, y, z], y] + 7 D[u[x, y, z], x] == 
     0) /. {x -> 0},
  (D[u[x, y, z], y] + D[v[x, y, z], x] == 0) /. {x -> 0},
  (D[u[x, y, z], z] + D[w[x, y, z], x] == 0) /. {x -> 0},
  (D[u[x, y, z], z] + D[w[x, y, z], x] == 0) /. {z -> 0},
  (D[v[x, y, z], z] + D[w[x, y, z], y] == 0) /. {z -> 0},
  (10500 D[w[x, y, z], z] + 4500 D[v[x, y, z], y] + 
      4500 D[u[x, y, z], x] == 13) /. z -> 0}

Trying to solve the problem results in an error message:

NDSolveValue[{pde, bcs}, {u, v, w}, {x, 0, 50}, {y, 0, 50}, {z, 0, 
  70}]

What is the problem here and how can it be fixed?

Attachments:

ndSolvePDE-direct.nb