### An Algebraic Composite Finite Element Mesh Method

#### Abstract

The composite ﬁnite element mesh method is useful for discretization error estimation and, in addition, for solution improvement with no increment in the computational cost. The technique consists in redeﬁne over a given mesh the linear operator that arises from the discretization of a partial diﬀerential equation. This operator is modiﬁed according to an appropriate linear combination between the operators of the given mesh and of a coarse mesh, which must be a coarsening of the ﬁrst one. In this work a novel algebraic composite mesh technique is proposed. The technique uses some tools from the Algebraic Multigrid method for the deﬁnition of the coarse mesh and the discrete space associated with it. Mesh coarsening is based on the fusion of elements in macroelements, with a new deﬁnition of the grid topology and basis functions. The agglomeration of elements is made in order to reduce the mesh anisotropy, which is of importance in the discretization of convection-diﬀusion-reaction problems. The discrete operator for the coarser mesh is obtained by the Galerkin Coarse Approximation, where inter-grid transfer operators are obtained using the graph of the coarse mesh. Several test problems with diﬀerent boundary conditions are presented.