The Proper Generalized Decomposition for Advanced Numerical Simulations

A separated representation of an error indicator for the mesh refinement process under the proper generalized decomposition framework

“…In first place, we show (in figure 12) the solution for two different values of the diffusion coefficient k. We observe how the smoothness of the solution decreases as k decreases. Figure 13 shows the local contributions to the error indicator for each one of the modes (Q x and Q y ) defined in (17) and (18). As in the previous example the error information is retained in the termê qx 1,r for the flux in x direction and inê qy 1,s for the y direction.…”

“…Thus, for example, in order to compute the solution of problem (1) for any value of the diffusion coefficient k ∈ Ω k , it suffices considering k as another coordinate (like x or y) and looking for u(x, y, k) as described in [17]. Thus, instead of (3) we will obtain the parametric solution:…”

“…Both functions (q x − q h x ) 2 and (q y − q h y ) 2 , after performing a postcompression using the separated representation constructor (see chapter 3 in [17]) can be written as…”

Coupled Systems in Mechanics: Fluid Structure Interactions

“…As it is well known, the equation governing such a problem is F = −kx which is coherent with (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17). Considering the dimension of each physical quantity, one can deduce the dimension of X and verify that it is well dimensionless:…”

“…Moreover, they are linear as the hypothesis of small motion was used. The 0-order approximation is given by equation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) and at the 1st-order by (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21), (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)…”