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Virtual Element Methods for hyperbolic problems on polygonal meshes
Giuseppe Vacca
TL;DR
The paper addresses applying VEM to hyperbolic problems on polygonal meshes, using linear wave equations as its model. It analyzes semi-discrete convergence and fully discrete Newmark and Bathe schemes; numerical tests report improved accuracy and no spurious oscillations with Bathe at small CLF.
Problem
Method
The paper discretizes wave equations with VEM in space, derives semi-discrete convergence estimates, and analyzes fully discrete Newmark and Bathe methods.
Source: §1 · p. 2, §3 · p. 3, §4 · p. 12Results
The Bathe method significantly improves solution accuracy and avoids spurious oscillations at small CLF, while Newmark trapezoidal exhibits them.
Source: §5 · p. 15Takeaways & Limitations
The paper presents VEM as a spatial discretization for wave equations and reports H1-seminorm and L2 convergence analysis alongside numerical tests.
Source: §1 · p. 2, §3 · p. 3With a non-stabilized bilinear form, the mass matrix may be singular and the maximum discrete eigenvalue unbounded, requiring care with Newmark β < 1/4.
Source: §5 · p. 14
Abstract
from arXiv · showhide
In the present paper we develop the Virtual Element Method for hyperbolic problems on polygonal meshes, considering the linear wave equations as our model problem. After presenting the semi-discrete scheme, we derive the convergence estimates in H^1 semi-norm and L^2 norm. Moreover we develop a theoretical analysis on the stability for the fully discrete problem by comparing the Newmark method and the Bathe method. Finally we show the practical behaviour of the proposed method through a large array of numerical tests.
1 Introduction
The introduction positions VEM as a polygonal-mesh approach and develops it for wave equations, with semi-discrete analysis, fully discrete schemes, and numerical tests. It also describes VEM’s computable polynomial projections and its prior use across multiple problem types.
1 Introduction
- VEM is described as a generalization of finite elements to polygonal or polyhedral meshes.The introduction also characterizes VEM as a recent technique for solving partial differential equations.Source: §1 · p. 1
- Polynomial projections make VEM bilinear forms computable from degrees of freedom using only polynomial integration, despite non-polynomial functions in the virtual spaces.The passage also states that the resulting method is conforming and recovers the correct order of convergence.Source: §1 · p. 1
- VEM satisfies the patch test: it gives the exact solution when that solution is a global polynomial of the selected accuracy degree.Source: §1 · p. 1
- Prior VEM applications include elasticity [8, 23], Stokes [17], Darcy, and time-dependent diffusion, alongside nonconforming [4] [22] and Serendipity [11] spaces.Source: §1 · pp. 1–2
- The paper develops VEM for wave equations, introducing discrete bilinear forms and analyzing semi-discrete error, fully discrete schemes, and numerical behavior.The fully discrete analysis considers Newmark and Bathe time integrators.Source: §1 · p. 2
2 The continuous problems
The continuous problem is a wave equation on a polygonal domain, with force and initial data specified under stated regularity assumptions. Its variational formulation uses the L2 scalar product, the grad-grad form, and a duality product.
2 The continuous problems
- The model is a second-order wave evolution problem on a polygonal domain Ω ⊂ R2.Source: §2 · p. 2
- The external force f belongs to L2(Ω × (0, T)), and the initial data u0 and z0 are assumed to belong to H1.Source: §2 · p. 2
- The variational formulation treats the second time derivative weakly and uses the L2 scalar product, grad-grad form, and force duality product.Source: §2 · p. 2
- The grad-grad form is continuous and coercive, and the stated problem has a unique solution.Source: §2 · p. 3
3 Virtual formulation of the wave equations
This section develops a polygonal-mesh Virtual Element discretization of the wave problem and states its semi-discrete formulation. It derives semi-discrete error estimates in the H¹ semi-norm and L² norm, including initial-error terms and contributions scaled by h^{k+1}.
3.1 Virtual spaces and bilinear forms
- The enhanced local space contains Pk(E), and its degrees of freedom comprise vertex values, values at k − 1 distinct points on each edge, and interior moments through order k − 2.Source: §3.1 · p. 4
- Under mesh assumptions A1–A3, the approximation results for Wh apply for 0 ≤ s ≤ k, with a constant independent of h.These results are presented as useful approximation results.Source: §3.1 · p. 5
- Computable discrete local bilinear forms approximate the continuous energy and L2 forms, with stability conditions that imply continuity of the global forms.The global approximated forms are assembled by summing their local contributions.Source: §3.1 · pp. 6–7
3.2 Virtual semi-discrete problems
- The semi-discrete virtual element problem uses the approximated mass and stiffness forms and has a unique solution for all t ∈ (0, T).The problem is posed against every test function in Wh and uses the approximated load term.Source: §3.2 · p. 7
3.3 Error analysis for the semi-discrete problems
- Under Theorem 3.3’s regularity assumptions, the semi-discrete error analysis derives an estimate and explicitly obtains an H1 semi-norm error bound.The stated assumptions include u ∈ C2(0, T; H1_0(Ω)) and u0, z0, ut, utt, and f(t) in H^{k+1}(Ω).Source: §3.3 · pp. 9–10
- Theorem 3.4 states that, under the assumptions of Theorem 3.3, its conclusion holds for every t in the interval (0,T).Source: §3.3 · p. 10
- The final estimate includes initial-error terms and h^{k+1}-scaled terms involving utt and f in L2(0,t; H^{k+1}(Ω)).The supplied passage is a fragment and does not preserve the full left-hand side or all norms.Source: §3.3 · p. 11
- Collecting estimates (27) and (41), with an argument similar to Theorem 3.3, gives the thesis.Source: §3.3 · p. 11
4 Fully discrete problems
The fully discrete analysis separates spatial and time-integration errors but focuses mainly on spatial discretization, examining Newmark and Bathe time integrators. It outlines their convergence and stability properties and discusses their behavior for wave propagation.
4 Fully discrete problems
- The fully discrete error has spatial and temporal components, and the analysis focuses mainly on spatial error while considering Newmark and Bathe time integration.The time-integrator contribution is expected to depend on the time step τ and the method's order p.Source: §4 · p. 12
- Newmark is at least first-order accurate; its second-order scheme is unconditionally stable for β ≥ 1/4, while 0 ≤ β < 1/4 requires a CFL time-step restriction.Second-order accuracy requires γ = 1/2.Source: §4 · p. 12
- The Bathe method is second-order and requires no analyst-selected parameter; it is reported stable in large-deformation and long-time responses where the trapezoidal rule fails.The paper notes that the Newmark trapezoidal rule can exhibit spurious oscillations, especially at high wave numbers, while Bathe is effective for wave propagation problems.Source: §4 · pp. 12–13
5 Numerical Tests
The numerical tests examine convergence of the VEM approximation and compare Bathe and Newmark time integration for wave propagation. They also test a non-stabilized bilinear form.
5 Numerical Tests
- The experiments assess H1-seminorm and L2 errors and compare Bathe with Newmark for wave propagation.The convergence test uses Newmark trapezoidal integration and VEM orders k = 1 and k = 2 on polygonal meshes.Source: §5 · p. 13
- The tests show the expected convergence order in h; for large h, the error is nearly constant in τ.Source: §5 · p. 14
- Errors from the original and reduced bilinear forms are very close, but the reduced form may yield a singular mass matrix and an unbounded maximum discrete eigenvalue, requiring care with Newmark β < 1/4.Source: §5 · p. 14
- The Bathe method significantly improves solution accuracy and avoids spurious oscillations, including for small CFL values.The comparison uses the Newmark trapezoidal rule and Bathe method with the VEM approximation.Source: §5 · p. 15