Matrix-oriented FEM formulation for stationary and time-dependent PDEs on x-normal domains

When numerical solution of elliptic and parabolic partial differential equations is required to be highly accurate in space, the discrete problem usually takes the form of large-scale and sparse linear systems. In this work, as an alternative, for spatial discretization we provide a Matrix-Oriented formulation of the classical Finite Element Method, called MO-FEM, of arbitrary order $k\in\mathbb{N}$. On structured 2D domains (e.g. squares or rectangles) the discrete problem is then reformulated as a Sylvester matrix equation, that we solve by the reduced approach in the associated spectral space. On a quite general class of domains, namely normal domains, and even on special surfaces, the MO-FEM yields a multiterm Sylvester matrix equation where the additional terms account for the geometric contribution of the domain shape. In particular, we obtain a sequence of these equations after time discretization of parabolic problems by the IMEX Euler method. We apply the matrix-oriented form of the Preconditioned Conjugate Gradient (MO-PCG) method to solve each multiterm Sylvester equation for MO-FEM of degree $k=1,\dots,4$ and for the lumped $\mathbb{P}_1$ case. We choose a matrix-oriented preconditioner with a single-term form that captures the spectral properties of the whole multiterm Sylvester operator. For several numerical examples, we show a gain in computational time and memory occupation wrt the classical vector approach solving large sparse linear systems by a direct method or by the vector PCG with same preconditioning. As an application, we show the advantages of the MO-FEM-PCG to approximate Turing patterns with high spatial resolution in a reaction-diffusion PDE system for battery modeling.