, we obtain for all j ? {2, J} ?? References

M. Ainsworth, A preconditioner based on domain decomposition for h-p finite-element approximation on quasi-uniform meshes, SIAM J. Numer. Anal, vol.33, pp.1358-1376, 1996.

P. F. Antonietti and G. Pennesi, V -cycle multigrid algorithms for discontinuous Galerkin methods on non-nested polytopic meshes, J. Sci. Comput, vol.78, pp.625-652, 2019.

P. F. Antonietti, M. Sarti, M. Verani, and L. T. Zikatanov, A uniform additive Schwarz preconditioner for high-order discontinuous Galerkin approximations of elliptic problems, J. Sci. Comput, vol.70, pp.608-630, 2017.

M. Arioli, E. H. Georgoulis, and D. Loghin, Stopping criteria for adaptive finite element solvers, SIAM, J. Sci. Comput, vol.35, pp.1537-1559, 2013.

I. Babu?ka, A. Craig, J. Mandel, and J. Pitkäranta, Efficient preconditioning for the p-version finite element method in two dimensions, SIAM J. Numer. Anal, vol.28, pp.624-661, 1991.

R. E. Bank, T. F. Dupont, and H. Yserentant, The hierarchical basis multigrid method, Numer. Math, vol.52, pp.427-458, 1988.

P. Bastian, M. Blatt, and R. Scheichl, Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems, Numer. Linear Algebra Appl, vol.19, pp.367-388, 2012.

R. Becker, C. Johnson, and R. Rannacher, Adaptive error control for multigrid finite element methods, Computing, vol.55, pp.271-288, 1995.

F. A. Bornemann and P. Deuflhard, The cascadic multigrid method for elliptic problems, Numer. Math, vol.75, pp.135-152, 1996.

L. Botti, A. Colombo, and F. Bassi, h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems, J. Comput. Phys, vol.347, pp.382-415, 2017.

J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp, vol.47, pp.103-134, 1986.

J. H. Bramble, J. E. Pasciak, and J. Xu, Parallel multilevel preconditioners, Numerical analysis 1989, vol.228, pp.23-39, 1989.

A. Brandt, S. Mccormick, and J. Ruge, Algebraic multigrid (AMG) for sparse matrix equations, Sparsity and its applications (Loughborough, 1983), pp.257-284, 1985.

S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, vol.15, 2008.

X. Cai and M. Sarkis, A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput, vol.21, pp.792-797, 1999.

L. Chen, R. H. Nochetto, and J. Xu, Optimal multilevel methods for graded bisection grids, Numer. Math, vol.120, pp.1-34, 2012.

P. G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, vol.4, 1978.

E. Efstathiou and M. J. Gander, Why restricted additive Schwarz converges faster than additive Schwarz, pp.945-959, 2003.

A. Ern and J. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol.159, 2004.

R. Eymard, T. Gallouët, and R. Herbin, Convergence of finite volume schemes for semilinear convection diffusion equations, Numer. Math, vol.82, pp.91-116, 1999.

S. Foresti, G. Brussino, S. Hassanzadeh, and V. Sonnad, Multilevel solution method for the p-version of finite elements, Computer Physics Communications, vol.53, pp.349-355, 1989.

A. Gholami, D. Malhotra, H. Sundar, G. Biros, F. Fft et al., A comparative study of state-of-the-art Poisson solvers for uniform and nonuniform grids in the unit cube, SIAM J. Sci. Comput, vol.38, pp.280-306, 2016.

M. Griebel, P. Oswald, and M. A. Schweitzer, A particle-partition of unity method. VI. A probust multilevel solver, Meshfree methods for partial differential equations II, vol.43, pp.71-92, 2005.

W. Hackbusch, of Springer Series in Computational Mathematics, vol.4, 2003.

R. Hiptmair, H. Wu, and W. Zheng, Uniform convergence of adaptive multigrid methods for elliptic problems and Maxwell's equations, Numer. Math. Theory Methods Appl, vol.5, pp.297-332, 2012.

B. Janssen and G. Kanschat, Adaptive multilevel methods with local smoothing for H 1 -and H curlconforming high order finite element methods, SIAM, J. Sci. Comput, vol.33, pp.2095-2114, 2011.

P. Jiránek, Z. Strako?, and M. Vohralík, A posteriori error estimates including algebraic error and stopping criteria for iterative solvers, SIAM J. Sci. Comput, vol.32, pp.1567-1590, 2010.

G. Kanschat, Robust smoothers for high-order discontinuous Galerkin discretizations of advectiondiffusion problems, J. Comput. Appl. Math, vol.218, pp.53-60, 2008.

M. Kronbichler and W. A. Wall, A performance comparison of continuous and discontinuous Galerkin methods with fast multigrid solvers, SIAM J. Sci. Comput, vol.40, pp.3423-3448, 2018.

S. Loisel, R. Nabben, and D. B. Szyld, On hybrid multigrid-Schwarz algorithms, J. Sci. Comput, vol.36, pp.165-175, 2008.

J. P. Lucero-lorca and G. Kanschat, Multilevel Schwarz preconditioners for singularly perturbed symmetric reaction-diffusion systems, 2018.

J. Mandel, Two-level domain decomposition preconditioning for the p-version finite element method in three dimensions, Internat. J. Numer. Methods Engrg, vol.29, pp.1095-1108, 1990.

D. Meidner, R. Rannacher, and J. Vihharev, Goal-oriented error control of the iterative solution of finite element equations, J. Numer. Math, vol.17, pp.143-172, 2009.

Y. Notay and A. Napov, A massively parallel solver for discrete Poisson-like problems, J. Comput. Phys, vol.281, pp.237-250, 2015.

P. Oswald, Multilevel finite element approximation, Teubner Skripten zur Numerik, Theory and applications, 1994.

J. Pape?, U. Rüde, M. Vohralík, and B. Wohlmuth, Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. HAL preprint 01662944, 2017.

J. Pape?, Z. Strako?, and M. Vohralík, Estimating and localizing the algebraic and total numerical errors using flux reconstructions, Numer. Math, vol.138, pp.681-721, 2018.

L. F. Pavarino, Additive Schwarz methods for the p-version finite element method, Numer. Math, vol.66, pp.493-515, 1994.

A. Quarteroni and G. Sacchi-landriani, Domain decomposition preconditioners for the spectral collocation method, J. Sci. Comput, vol.3, pp.45-76, 1988.

J. Schöberl, J. M. Melenk, C. Pechstein, and S. Zaglmayr, Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements, IMA J. Numer. Anal, vol.28, pp.1-24, 2008.

J. Schöberl, C++11 Implementation of Finite Elements in NGsolve, tech. rep., ASC Report 30/2014, Institute for Analysis and Scientific Computing, 2014.

H. Sundar, G. Stadler, and G. Biros, Comparison of multigrid algorithms for high-order continuous finite element discretizations, Numer, Linear Algebra Appl, vol.22, pp.664-680, 2015.

M. Vohralík, On the discrete Poincaré-Friedrichs inequalities for nonconforming approximations of the Sobolev space H 1, Numer. Funct. Anal. Optim, vol.26, pp.925-952, 2005.

T. Warburton, An explicit construction of interpolation nodes on the simplex, J. Engrg. Math, vol.56, pp.247-262, 2006.

H. Wu and Z. Chen, Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems, Sci. China Ser. A, vol.49, pp.1405-1429, 2006.

J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev, vol.34, pp.581-613, 1992.

J. Xu, L. Chen, and R. H. Nochetto, Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids, Multiscale, nonlinear and adaptive approximation, pp.599-659, 2009.

X. Zhang, Multilevel Schwarz methods, Numer. Math, vol.63, pp.521-539, 1992.