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

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