A Weak Galerkin Finite Element Method for the Maxwell Equations

Lin Mu; Junping Wang; Xiu Ye; Shangyou Zhang

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A Weak Galerkin Finite Element Method for the Maxwell Equations

Lin Mu, Junping Wang, Xiu Ye, Shangyou Zhang

arXiv:1312.2309v1Dec 9, 2013math.NA

TL;DR

The paper addresses numerical approximation of time-harmonic Maxwell equations using a flexible weak Galerkin framework. It combines discrete weak curl and gradient operators with stabilization and boundary-variable reduction, and reports optimal-order convergence with numerical confirmation. The method is analyzed on shape-regular polyhedral meshes, while one simpler element choice has numerical but not theoretical support.

Problem
The paper seeks a numerical method for time-harmonic Maxwell equations that accommodates discontinuous functions on arbitrary-shape polyhedral meshes.
Method
The method combines discrete weak curl and gradient operators with stabilization, then eliminates element-interior variables to obtain a boundary-unknown system.
Results
The WG approximations achieve optimal-order error estimates in appropriate discrete norms, and numerical tests confirm the theoretical convergence estimates.
Takeaways & Limitations
The method provides a parameter-independent WG scheme with fewer globally coupled unknowns than DG methods and flexibility on arbitrary polygonal or polyhedral partitions.
Takeaways & Limitations
For the choice s = t = k and ℓ = ι = k − 1, the paper reports numerical experiments but leaves theoretical analysis to future investigation.

Abstract

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This paper introduces a numerical scheme for time harmonic Maxwell's equations by using weak Galerkin (WG) finite element methods. The WG finite element method is based on two operators: discrete weak curl and discrete weak gradient, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on arbitrary shape of polyhedra and, at the same time, is parameter free. Optimal-order of convergence is established for the weak Galerkin approximations in various discrete norms which are either $H^1$-like or $L^2$ and $L^2$-like. An effective implementation of the WG method is developed through variable reduction by following a Schur-complement approach, yielding a system of linear equations involving unknowns associated with element boundaries only. Numerical results are presented to confirm the theory of convergence.

1. Introduction.

The paper develops a weak Galerkin method for time-harmonic Maxwell equations, combining discrete weak curl and gradient operators. The method supports arbitrary polygonal or polyhedral partitions, reduces globally coupled unknowns through boundary variables, and establishes optimal-order error estimates.

Method: The WG method combines a discrete curl operator with a discrete weak gradient to form a finite element scheme for Maxwell equations.Two-component weak functions represent interior and element-boundary values.
Convergence: Optimal-order error estimates are established for the WG approximations in appropriate discrete norms when s = t = k, ℓ = k − 1, and ι = k.For ℓ = ι = k − 1, numerical experiments are provided without theoretical analysis.
Flexibility: Weak functions and weak derivatives permit finite element construction on partitions containing arbitrary polygons or polyhedrons.The method is parameter independent in stability and convergence.
Implementation: Eliminating element-interior variables u0 and p0 locally yields a linear system involving only boundary variables ub and pb.This produces fewer globally coupled unknowns than the compared DG formulation.
Organization: The paper derives discrete weak differential operators, analyzes approximation errors, and presents numerical results intended to verify the established theory.The implementation uses variable reduction or elimination, while the final section reports numerical results.

2. Preliminaries and Notations.

This section introduces notation for multivariable derivatives, Sobolev norms and inner products, and the H(curl) space used for Maxwell analysis. It also motivates weak analogues of curl and gradient for discontinuous functions.

Notation: The preliminaries define multi-index notation through α, its total order |α|, and the differential operator ∂α.
Function spaces: H(curl; D) consists of vector-valued functions whose values and curls are square integrable.
Weak operators: The paper introduces weak curl and weak gradient operators as analogues of curl and gradient for discontinuous applied functions.

3. Weak Derivatives.

The paper represents functions by separate interior and boundary components, then defines weak and discrete weak gradient and curl operators through elementwise duality relations. These constructions support discontinuous functions on polyhedral domains.

Weak functions: A weak function on a polyhedral region is v = {v0, vb}, with an interior component v0 and an independent boundary component vb.The boundary component need not be the trace of the interior component.
Weak gradient: The weak gradient is defined as a linear functional through an interior divergence term and a boundary normal-trace pairing.
Weak gradient: The discrete weak gradient is the unique polynomial satisfying the corresponding relation for all polynomial test vectors.
Weak curl: Vector-valued weak functions use v0 in the element and vb × n on the boundary to define weak curl.
Weak curl: The weak curl and discrete weak curl are defined through interior curl pairings and boundary pairings with vb × n.
Representation: The boundary vector can be restricted to tangential components because only vb × n enters the formulation, reducing unknowns.
Meshes: Shape-regular meshes may consist of polyhedra of arbitrary shape.

4. Numerical Algorithms.

The WG algorithm uses elementwise polynomial interior and face spaces, discrete weak curl and gradient operators, and stabilization terms. Its unique solution can be computed through local variable elimination and boundary conditions.

Finite element spaces: The finite element spaces use degree-k vector polynomials in elements and on faces, with degree-(k − 1) scalar interior polynomials and degree-k face polynomials.
Discrete operators: Discrete weak gradient and curl are computed elementwise using the defining relations on the finite element spaces.
Stabilization: The bilinear form for the vector component combines the weak-curl term with a stabilization term.
Algorithm: The WG algorithm imposes the boundary condition through ub × n = Qbφ and couples the vector and scalar unknowns in two variational equations.
Well-posedness: The WG finite element algorithm has a unique solution.
Well-posedness: The uniqueness proof uses vanishing weak curl, divergence, normal jumps, and boundary conditions to show that the homogeneous solution is zero.

5. Error Equations.

This section establishes projection identities for the discrete weak curl and gradient, then derives error equations for the WG approximation using integration by parts and stabilization terms.

Projection operators: The projections Qh and Qh are defined locally to map exact vector and scalar fields into the WG finite element spaces.Qb is the L2 projection onto Pk(e), while Qh includes interior and boundary components.
Projection identities: The discrete weak gradient commutes with projection: ∇w(Qhq) = Q0(∇q) for every q ∈ H1(Ω).This identity follows from integration by parts and the definitions of the projections and discrete gradient.
Projection identities: The projected discrete weak curl satisfies an analogous relation obtained by integration by parts and testing against polynomial vector fields.The argument identifies the projected exact curl with the discrete weak-curl expression.
Error equations: Subtracting the WG equations from the projected equations yields the error equations governing the vector and scalar approximation errors.The error functions are defined relative to the L2 projections of the exact solution.
Error equations: Substituting projected exact solutions into the WG scheme produces consistency terms ϕu and φu,p in the two discrete equations.The resulting relations are a(Qhu, v) − b(v, Qhp) = (f, v0) + ϕu(v) and b(Qhu, q) + s2(Qhp, q) = −(g, q0) + φu,p(q).

6. Preparation for Error Estimates.

This section defines discrete norms and establishes projection and consistency estimates needed to analyze WG errors on general polyhedral meshes.

Discrete norms: The WG vector seminorm is strengthened with additional terms to obtain a norm on Vh,0.One added term measures normal jumps of element-interior functions across edges or faces.
Discrete norms: The scalar space Wh is equipped with a discrete norm used in the subsequent inf-sup and error estimates.The section introduces this norm for arbitrary q ∈ Wh.
Projection estimates: Projection estimates are established for Qh, Qh, and Qh on WG shape-regular partitions, with regularity indices satisfying 0 ≤ t ≤ k and 0 ≤ s ≤ 1.These estimates support approximation control for both interior and boundary components.
Technical estimates: Trace and inverse inequalities are used for arbitrary polyhedra to control boundary terms involving polynomial functions.The trace inequality applies on an element face, and the polynomial case yields a standard inverse inequality.
Consistency estimates: Consistency terms are bounded by approximation factors involving h^t and discrete norms of the test functions.For example, |s2(Qhp, q)| + |l2(w, q)| is bounded by C h^t(∥w∥t+1 + ∥p∥t)|q|0,h.

7. Error Estimates.

The section derives optimal-order WG error estimates in discrete norms by combining the error equations, inf-sup arguments, consistency bounds, and jump controls.

Discrete-norm estimates: Lemma 7.1 constructs a test function vq from each q ∈ Wh to support control of the scalar-component error.The construction uses the discrete gradient of q0 and vanishing boundary component.
Main error estimates: Theorem 7.2 combines the two error equations with the consistency estimates to obtain bounds for the WG approximation under 1/2 < t ≤ k.The analysis assumes u ∈ [H^{t+1}(Ω)]3 and p ∈ H^t(Ω).
Consistency bounds: The consistency functional satisfies |φu,p(ϵh)| ≤ C h^t(∥u∥t+1 + ∥p∥t)|ϵh|0,h.This estimate is used with the error equations to derive scalar error bounds.
Full discrete norm: Additional estimates control the normal-jump and divergence-related terms required to upgrade the vector seminorm to the full discrete norm.The proof uses specially chosen q functions and trace inequalities.
Summary estimate: Theorem 7.4 summarizes the resulting error estimate for the WG finite element approximations.The theorem is stated under the assumptions of Theorem 7.2.

8. An Error Estimate in L2.

This section derives an L2 error estimate for the vector component through an auxiliary regularity problem and projection-based consistency bounds.

Auxiliary problem: The auxiliary problem assumes [H1+s(Ω)]3 × Hs(Ω) regularity for its solution, with 0 < s ≤ 1.This regularity property provides the estimate needed in the duality argument.
Duality argument: The L2 analysis tests the auxiliary problem with the vector approximation error and uses the WG error equations.The proof introduces projected auxiliary solutions and combines several consistency relations.
Consistency estimate: The consistency functional is bounded by C h^r(∥u∥r+1 + ∥p∥r)(|Qhψ|1,h + |Qhξ|0,h).This estimate controls the terms arising in the auxiliary-problem argument.
Conclusion: The final substitution of the intermediate estimates yields the desired L2 error estimate for the vector component.The proof concludes after applying the regularity estimate and projection bounds.

9. An Effective Implementation through Variable Reduction.

The WG scheme is implemented efficiently by eliminating element-interior variables locally and solving a Schur-complement system for boundary variables. The resulting method reduces globally coupled unknowns while retaining local recovery of interior components.

Local WG formulation: Because the testing functions are locally supported, equations (9.1)–(9.2) reduce to elementwise problems for the interior variables u0 and p0.The local equations include (9.5) for v = {v0, 0} and (9.6) for q = {q0, 0}.
Variable reduction: Given boundary variables ub and pb, the interior variables u0 and p0 are recovered independently on each element through local solves.The recovery operators are written as u0 = D(ub, pb, f, g) and p0 = E(ub, pb, f, g).
Variable reduction: Superposition separates the interior solutions into contributions from boundary variables and forcing data, enabling substitution into the remaining boundary equations.The operators are decomposed into D1, D2, E1, and E2 before substitution into the global system.
Schur-complement implementation: The reduced square system has ub and pb as unknowns; after solving it, u0 and p0 are recovered locally, forming the WG Schur complement implementation.This reduces the number of globally coupled unknowns significantly for practical implementation.
Numerical setup: Numerical tests use first-order WG elements on successively refined cubic meshes, with piecewise linear vector components and mixed constant-linear scalar components.The experiments are conducted on the unit cube, refining each cube into eight half-sized cubes.

10. Numerical Results.

The numerical tests verify the WG convergence theory across several exact solutions, while also revealing superconvergence and excellent performance for a lower-order variant without established theory.

The first test produced zero errors for the linear and constant exact solutions, up to computer accuracy.
The second test reproduced p numerically and achieved a half-order-higher convergence rate for u than proved theoretically.The observed superconvergence is attributed probably to the special format of the exact solution.
The third test with a general exact solution confirmed the theoretical convergence estimates in Theorems 7.4 and 8.1.
The discrete scalar L2 seminorm measures face-center errors and converged at O(h2), higher than the theoretical prediction.The authors suggest that superconvergence may explain this behavior.
The fourth test reconfirmed earlier convergence results, with numerical performance similar to the third test.
A lower-order WG variant showed excellent approximation, although the paper provides no convergence theory for this element.It uses piecewise linear vector functions and piecewise constant scalar functions on element interiors and boundaries.