This paper extends an algorithm of P^{1}-conservative interpolation on triangular
meshes to tetrahedral meshes and thus constructs an approach of solution reconstruction
for three-dimensional problems. The conservation property is achieved
by local mesh intersection and the mass of a tetrahedron of the current mesh is calculated
by the integral on its intersection with the background mesh. For each current
tetrahedron, the overlapped background tetrahedrons are detected efficiently. A mesh
intersection algorithm is proposed to construct the intersection of a current tetrahedron
with the overlapped background tetrahedron and mesh the intersection region
by tetrahedrons. A localization algorithm is employed to search the host units in background
mesh for each vertex of the current mesh. In order to enforce the maximum
principle and avoid the loss of monotonicity, correction of nodal interpolated solution
on tetrahedral meshes is given. The performance of the present solution reconstruction
method is verified by numerical experiments on several analytic functions and the
solution of the flow around a sphere.