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Virtual element method for semilinear elliptic problems on polygonal meshes. (English) Zbl 1433.65271

The paper is concerned with the virtual element method for the numerical solution of the semilinear elliptic problem \[ -\nabla \cdot \left( G \left( x \right) \nabla u \left( x \right) \right) = f \left( u \left( x \right), x \right)\] in a two-dimensional convex polygonal domain. The equation is equipped with homogeneous Dirichlet boundary conditions and it is assumed that the function \(f\) is Lipschitz continuous with respect to the first variable. The standard weak formulation is derived and the virtual element method is proposed for its numerical solution. The method consists in using arbitrary polytopal elements satisfying star convexity condition rather than the triangles or quadrilaterals as in the standard finite element method. The main feature of the method is that the local shape function space in each element is defined implicitly. The approximation of the nonlinear term based on the \(L^2\) projection is proposed. The existence and uniqueness of the discrete solution are proven and optimal a priori \(H^1\) error estimates are derived. The efficiency and applicability of the method are illustrated on two numerical examples where the optimal convergence rate with respect to the \(L^2\)-norm as well as the \(H^1\)-norm is achieved.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J61 Semilinear elliptic equations

Software:

PolyMesher
PDFBibTeX XMLCite
Full Text: DOI

References:

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