Boundary observability for the finite-difference space semi-discretizations of the 2-d wave equation in the square

AbstractWe extend our previous results on the boundary observability of the finite-difference space semidiscretizations of the 1-d wave equation to 2-d in the square. As in the 1-d case, we prove that the constants on the boundary observability inequality blow-up as the mesh-size tends to zero. However, we prove a uniform observability inequality in a subspace of solutions generated by the low frequencies. The dimension of these subspaces grows as the mesh size tends to zero and eventually, in the limit, covers the whole energy space. Our result is sharp in the sense that the uniformity of the observability inequality is lost when the dimension of the subspaces grows faster. Our method of proof combines discrete multiplier techniques, Fourier series developments and compactness-uniqueness arguments