The addition of fields on different meshes

Let T1T1 and T2T2 be two (arbitrarily unrelated) meshes of the same domain Ω⊂Rd,1⩽d⩽3Ω⊂Rd,1⩽d⩽3. Let V1V1 and V2V2 be the discrete function spaces induced from these meshes by equipping them with some choice of basis functions. This work discusses the following problem: given two functions f1∈Vff1∈Vf and f2∈Vgf2∈Vg, compute f1 + f2. Despite the simple statement of the task, the general solution is not obvious, for the desired sum lies neither in V1V1 nor in V2V2, and is not representable on either mesh. The problem is solved by efficiently constructing a supermesh  TSTS of T1T1 and T2T2 which induces a superspace of both V1V1 and V2V2. Once such a mesh is available, the desired quantity may be represented exactly (subject to roundoff).