A Non-perturbative Approach to Computing Seismic Normal Modes in Rotating Planets

Abstract

A continuous Galerkin method based approach is presented to compute the seismic normal modes of rotating planets. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core using a polynomial filtering eigensolver. The relevant elastic-gravitational system of equations, including the Coriolis force, is subjected to a mixed finite-element method, while self-gravitation is accounted for with the fast multipole method. Our discretization utilizes fully unstructured tetrahedral meshes for both solid and fluid regions. The relevant eigenvalue problem is solved by a combination of several highly parallel and computationally efficient methods. We validate our three-dimensional results in the non-rotating case using analytical results for constant elastic balls, as well as numerical results for an isotropic Earth model from standard “radial” algorithms. We also validate the computations in the rotating case, but only in the slowly-rotating regime where perturbation theory applies, because no other independent algorithms are available in the general case. The algorithm and code are used to compute the point spectra of eigenfrequencies in several Earth and Mars models studying the effects of heterogeneity on a large range of scales.

Appendices

Construction of Orthonormal Bases and Submatrices

Here, we introduce three-dimensional polynomial bases $\{ \psi _n^s \}_{n=1}^{N_{p^s}}$, $\{ \psi _n^f \}_{n=1}^{N_{p^f}}$ and $\{ \psi _n^p \}_{n=1}^{N_{p^p}}$ while addressing the fact that the Lagrange polynomials are not orthogonal to one another. We suppress superscripts s, f, p in the notation in the remainder of this subsection. To simplify the computations, we introduce reference volume and boundary elements. That is, we introduce a mapping that connects any element K to the reference tetrahedron defined by

$$\begin{aligned} {\mathbf {I}} = \{ r = (r_1,r_2,r_3)\ :\ r_1 \ge -1 ,\ r_2 \ge -1 ,\ r_3 \ge -1,\ r_1 + r_2 + r_3 \le -1 \} . \end{aligned}$$

Likewise, we introduce a mapping that connects any boundary element E to the reference triangle defined by

$$\begin{aligned} {\mathbf {I}}_{2D} = \{ t = (t_1,t_2)\ :\ t_1 \ge -1 ,\ t_2 \ge -1 ,\ t_1 + t_2 \le 0 \} . \end{aligned}$$

We note that any two tetrahedra are connected through an affine transformation, $x \rightarrow r$, with a constant Jacobian, J, which is the determinant of $(\partial _r x)$. For the local approximation on the reference element ${\mathbf {I}}$, we have

$$\begin{aligned} u_j(r) = \sum _{n=1}^{N_p} ({\hat{u}}_j)_n \psi _n(r) = \sum _{i=1}^{N_p} u_j(r_i) \ell _i(r) . \end{aligned}$$

The vector fields are treated component-wise in our discretization. This yields the expression ${\mathcal {V}} {\hat{u}}_j = u_j$, where the generalized Vandermonde matrix takes the form of ${\mathcal {V}}_{in} = \psi _n(r_i)$ with i, n as indices of nodal points. Here, $\{\psi _n \}$ is a polynomial basis that is orthonormal on ${\mathbf {I}}$. We later introduce submatrices of ${\mathcal {V}}$. We then evaluate derivatives and mass matrices according to

$$\begin{aligned} \partial _{x_i} = (\partial _{x_i} r_j) {\mathcal {D}}_{j} , \quad {\mathcal {D}}_{j} = (\partial _{r_j} {\mathcal {V}}) {\mathcal {V}}^{-1} , \quad {\mathcal {M}} = {\mathcal {V}}^{-T} {\mathcal {V}}^{-1} , \end{aligned}$$

where ${\mathcal {D}}_j$ and ${\mathcal {M}}$ are the derivative matrix and the mass matrix on the reference tetrahedron. More details of the constructions of J, ${\mathcal {V}}$, ${\mathcal {D}}_j$ and ${\mathcal {M}}$ can be found in [63, Chapter 10.1]. Thus, we introduce

$$\begin{aligned} {\mathcal {V}}_s ,\ {\mathcal {V}}_f ,\ {\mathcal {V}}_p, \quad {\mathcal {M}}_s ,\ {\mathcal {M}}_f ,\ {\mathcal {M}}_p \quad \text {and}\quad {\mathcal {D}}_j^s ,\ {\mathcal {D}}_j^f ,\ {\mathcal {D}}_j^p. \end{aligned}$$

We employ the notation

$$\begin{aligned} {\mathsf {D}}^s_i = (\partial _{x_i} r_j) {\mathcal {D}}^s_j ,\quad {\mathsf {D}}^f_i = (\partial _{x_i} r_j) {\mathcal {D}}^f_j ,\quad {\mathsf {D}}^p_i = (\partial _{x_i} r_j) {\mathcal {D}}^p_j , \end{aligned}$$

reflecting the mapping of the derivatives from the reference tetrahedron to the target element. We follow a similar approach for boundary elements and introduce

$$\begin{aligned} {\mathcal {M}}_s^{2D}, \ {\mathcal {M}}_f^{2D} \quad \text {and}\quad J^{2D}, \end{aligned}$$

where ${\mathcal {M}}_s^{2D}$ and ${\mathcal {M}}_f^{2D}$ are the mass matrices for solid and fluid boundary elements, respectively; $J^{2D}$ denotes the Jacobian, which is the determinant of $(\partial _t x)$ on the boundary element. The construction of the mass matrices ${\mathcal {M}}^{2D}_s$ and ${\mathcal {M}}^{2D}_f$ on the reference triangle ${\mathbf {I}}_{2D}$ is similar to the construction of ${\mathcal {M}}$ [63, Chapter 6.1].

1.1 Submatrices: $A_{sg}$, $A_f$, $A_p$, $M_s$, $M_f$, $R_s$ and $R_f$

We extract ${\tilde{u}}^s |_{K_k}$, ${\tilde{u}}^f |_{K_k}$ and ${\tilde{p}} |_{K_k}$ from ${\tilde{u}}^s$, ${\tilde{u}}^f$ and ${\tilde{p}}$, respectively, by restricting the nodes to the ones of element $K_k$. In a similar fashion, we extract ${\tilde{v}}^s |_{K_k}$, ${\tilde{v}}^f |_{K_k}$ and ${\tilde{v}}^p |_{K_k}$ on any element $K_k$. For the evaluation of matrix $A_{sg}$ in Table 3 we need to evaluate the submatrices on element $K_k$ through

$$\begin{aligned} \int _{K_k^{\text {S}}} \partial _{x_i} ({\overline{v}}^s_h)_j (c_{ijmn} \partial _{x_m} (u^s_h)_n) \,\mathrm {d}x&= ({\tilde{v}}^s_j|_{K_k})^{{\mathsf {H}}}[ J_k ({\mathsf {D}}_i^s)^{{\mathsf {T}}}c^{k}_{ijmn} {\mathcal {M}}_s {\mathsf {D}}_m^s ] {\tilde{u}}^s_n|_{K_k} , \end{aligned}$$

(61)

$$\begin{aligned} \int _{K_k^{\text {S}}} \partial _{x_i} ({\overline{v}}^s_h)_i g'_j (u^s_h)_j \rho ^0 \,\mathrm {d}x&= ({\tilde{v}}^s_i|_{K_k})^{{\mathsf {H}}}[ J_k ({\mathsf {D}}_i^s)^{{\mathsf {T}}}\rho ^0_k {\mathcal {M}}_s D_{g'_j} ] {\tilde{u}}^s_j|_{K_k} , \end{aligned}$$

(62)

$$\begin{aligned} \int _{K_k^{\text {S}}} - (u^s_h)_i \partial _{x_i} g'_j ({\overline{v}}^s_h)_j \rho ^0 \,\mathrm {d}x&= ({\tilde{v}}^s_i|_{K_k})^{{\mathsf {H}}}[- J_k \rho ^0_k D_{\partial _{x_i} g'_j} {\mathcal {M}}_s ] {\tilde{u}}^s_j|_{K_k} , \end{aligned}$$

(63)

$$\begin{aligned} \int _{K_k^{\text {S}}} - (u^s_h)_j (\partial _{x_j} ({\overline{v}}^s_h)_i) g'_i \rho ^0 \,\mathrm {d}x&= ({\tilde{v}}^s_i|_{K_k})^{{\mathsf {H}}}[- J_k {\mathsf {D}}^s_j {\mathcal {M}}_s \rho ^0_k D_{g'_i} ] {\tilde{u}}^s_j|_{K_k} , \end{aligned}$$

(64)

where $c^{k}_{ijmn}$, $\rho ^0_k$ and $J_k$ denote the stiffness tensor, density and the Jacobian on element $K_k$, respectively; $D_{g'_i}$ and $D_{\partial _{x_i} g'_j}$ denote the diagonal matrices whose diagonal entries are $g'_i$ and $\partial _{x_i} g'_j$, respectively. For the evaluation of the boundary integration in $A_{sg}$, we need to evaluate the submatrix on element $E_l^{\text {FS}}$ through

$$\begin{aligned} \int _{E_l^{\text {FS}}} ({\overline{v}}^s_h)_i g'_i \nu ^{s \rightarrow f}_j (u^s_h)_j [\rho ^0]^f \,\mathrm {d}\Sigma = ({\tilde{v}}^s_i|_{E_l})^{{\mathsf {H}}}[ J^{2D}_l \rho ^0_l D_{g'_i} {\mathcal {M}}_s^{2D} \nu ^{s \rightarrow f}_j|_{E_l} ] {\tilde{u}}^s_j|_{E_l} , \end{aligned}$$

(65)

where $\rho ^0_l$ and $\nu ^{s \rightarrow f}_j|_{E_l}$ denote the density and normal vector on the boundary element $E_l^{\text {FS}}$, respectively, upon extracting ${\tilde{v}}^s_i |_{E_l}$ and ${\tilde{u}}^s_i|_{E_l}$. We can deal with the integral over $\Sigma ^{\text {FF}}$ similarly.

We then evaluate the submatrices for $A_f$, $A_p$, $M_s$, $M_f$ in Table 3 and obtain

$$\begin{aligned} \int _{K_k^{\text {F}}} \rho ^0 N^2 \frac{ g'_i ({\overline{v}}^f_h)_i g'_j (u^f_h)_j }{\Vert g' \Vert ^2} \,\mathrm {d}x&= ({\tilde{v}}^f_i|_{K_k})^{{\mathsf {H}}}[ J_k D_{g'_i/\Vert g'\Vert } \rho _k^0 N_k^2 {\mathcal {M}}_f D_{g'_j/\Vert g'\Vert } ] {\tilde{u}}^f_j|_{K_k} , \end{aligned}$$

(66)

$$\begin{aligned} \int _{K_k^{\text {F}}} - {\overline{v}}^p_h p_h \kappa ^{-1} \,\mathrm {d}x&= ({\tilde{v}}^p|_{K_k})^{{\mathsf {H}}}[ - J_k \kappa ^{-1}_k {\mathcal {M}}_p ] {\tilde{p}}|_{K_k} , \end{aligned}$$

(67)

$$\begin{aligned} \int _{K_k^{\text {S}}} ({\overline{v}}^s_h)_i (u^s_h)_i\rho ^0 \,\mathrm {d}x&= ({\tilde{v}}^s_i|_{K_k})^{{\mathsf {H}}}[ J_k \rho ^0_k {\mathcal {M}}_s ] {\tilde{u}}^s_i|_{K_k} , \end{aligned}$$

(68)

$$\begin{aligned} \int _{K_k^{\text {F}}} ({\overline{v}}^f_h)_i (u^f_h)_i \rho ^0 \,\mathrm {d}x&= ({\tilde{v}}^f_i|_{K_k})^{{\mathsf {H}}}[ J_k \rho ^0_k {\mathcal {M}}_f ] {\tilde{u}}^f_i|_{K_k} , \end{aligned}$$

(69)

where $D_{g'_j/\Vert g'\Vert }$ denotes a diagonal matrix whose diagonal entries are $g'_j/\Vert g'\Vert$ and $N_k^2$ denotes the square of the Brunt-Väisälä frequency on element $K_k$. We also obtain the rotation components $R_s$ and $R_f$,

$$\begin{aligned} \int _{K_k^{\text {S}}} \epsilon _{ijm} ({\overline{v}}^s_h)_i (u^s_h)_j \rho ^0 \,\mathrm {d}x&= ({\tilde{v}}^s_i|_{K_k})^{{\mathsf {H}}}[ \epsilon _{ijm} J_k \rho ^0_k {\mathcal {M}}_s ] {\tilde{u}}^s_j|_{K_k} , \end{aligned}$$

(70)

$$\begin{aligned} \int _{K_k^{\text {F}}} \epsilon _{ijm} ({\overline{v}}^f_h)_i (u^f_h)_j \rho ^0 \,\mathrm {d}x&= ({\tilde{v}}^f_i|_{K_k})^{{\mathsf {H}}}[ \epsilon _{ijm} J_k \rho ^0_k {\mathcal {M}}_f ] {\tilde{u}}^f_j|_{K_k} , \end{aligned}$$

(71)

where $\epsilon _{ilm}$ denotes the Levi-Civita symbol.

1.2 Submatrices: $A_{\text {dg}}$ and $A_{\text {dg}}^{{\mathsf {T}}}$

Here, we discuss the integration between the different variables. For the inner products between $u^f_h$ and $p_h$ for $A_{\text {dg}}$ and $A_{\text {dg}}^{{\mathsf {T}}}$ in Table 3, we evaluate the mass matrices ${\mathcal {M}}_{pf}$ and ${\mathcal {M}}_{fp}$,

$$\begin{aligned} {\mathcal {M}}_{pf} = ({\mathcal {V}}_p^{-1}(I_{f}))^T {\mathcal {V}}_{f}^{-1}(I_{p}) , \quad {\mathcal {M}}_{fp} = ({\mathcal {V}}_{f}^{-1}(I_{p}))^T {\mathcal {V}}_p^{-1}(I_{f}) , \end{aligned}$$

where we refine the notation to indicate submatrices of ${\mathcal {V}}$; ${\mathcal {V}}(I)$ denotes the submatrix of ${\mathcal {V}}$ formed by columns indexed by $I \subseteq \{1,\ldots , N_{p}\}$. The selection of submatrices is based on the polynomial construction [63, (10.6)]. For instance, if the polynomial orders used for both $u^f_h$ and $p_h$ are the same, i.e., $p^f=p^p$, $I_{f}=I_{p}=\{1,\ldots , N_{p^f}\}$; if $p^p=1$ and $p^f=2$, we have $N_{p^p}=4$, $N_{p^f}=10$ and $I_{f}=\{1,2,3,4\}$, $I_{p}=\{1,2,4,7\}$. It is apparent that ${\mathcal {M}}_{pf} = {\mathcal {M}}_{fp}^T$.

Evaluating $A_{\text {dg}}$ in Table 3 requires the evaluation of the submatrices on element $K_k$ through

$$\begin{aligned} \int _{K_k^{\text {F}}} ({\overline{v}}^f_h)_j (\partial _{x_j} p_h ) \,\mathrm {d}x&= ({\tilde{v}}^f_j|_{K_k})^{{\mathsf {H}}}[ J_k {\mathcal {M}}_{fp} {\mathsf {D}}_{j}^p ] {\tilde{p}}|_{K_k} , \end{aligned}$$

(72)

$$\begin{aligned} \int _{K_k^{\text {F}}} ({\overline{v}}^f_h)_j g'_j p_h \rho ^0 \kappa ^{-1}\,\mathrm {d}x&= ({\tilde{v}}^f_j|_{K_k})^{{\mathsf {H}}}[ J_k D_{g'_j} \rho ^0_k \kappa ^{-1}_k {\mathcal {M}}_{fp} ] {\tilde{p}}|_{K_k} , \end{aligned}$$

(73)

where $\kappa ^{-1}_k$ denotes the inverse of the bulk modulus on element $K_k$. To evaluate $A_{\text {dg}}^{{\mathsf {T}}}$ in Table 3, we also need to evaluate the submatrices on element $K_k$ through

$$\begin{aligned} \int _{K_k^{\text {F}}} (\partial _{x_j} {\overline{v}}^p_h) (u^f_h)_j \,\mathrm {d}x&= ({\tilde{v}}^p|_{K_k})^{{\mathsf {H}}}[ J_k ({\mathsf {D}}_{j}^p)^{T} {\mathcal {M}}_{pf} ] {\tilde{u}}^f_j|_{K_k} , \end{aligned}$$

(74)

$$\begin{aligned} \int _{K_k^{\text {F}}} {\overline{v}}^p_h g'_j (u^f_h)_j \rho ^0 \kappa ^{-1} \,\mathrm {d}x&= ({\tilde{v}}^p|_{K_k})^{{\mathsf {H}}}[ J_k \rho ^0_k \kappa ^{-1}_k {\mathcal {M}}_{pf} D_{g'_j} ] {\tilde{u}}^f_j|_{K_k} . \end{aligned}$$

(75)

1.3 Submatrices: $E_{\text {FS}}$ and $E_{\text {FS}}^{{\mathsf {T}}}$

For $E_{\text {FS}}$ and $E_{\text {FS}}^{{\mathsf {T}}}$ , similar to Sect. A.2, we introduce two new indices to construct ${\mathcal {M}}_{ps}^{2D}$ and ${\mathcal {M}}_{sp}^{2D}$ on the boundary elements associated with the fluid–solid boundary. The selection of the submatrix is based on [63, Chapter 6]. ${\mathcal {M}}_{ps}^{2D} = {{\mathcal {M}}_{sp}^{2D}}^{{\mathsf {T}}}$ holds true as well. To evaluate $E_{\text {FS}}^{{\mathsf {T}}}$ in Table 3, we need to compute the submatrix on boundary element $E^{\text {FS}}_l$ through

$$\begin{aligned} \int _{E_l^{{\text {FS}}}} ({\overline{v}}^s_h)_j \nu ^{s \rightarrow f}_j p_h \,\mathrm {d}\Sigma = ( {\tilde{v}}^s_j|_{E_l} ) ^{{\mathsf {H}}}[ J^{2D}_l \nu ^{s \rightarrow f}_j {\mathcal {M}}_{sp}^{2D} ] {\tilde{p}}|_{E_l} , \end{aligned}$$

(76)

upon extracting ${\tilde{p}}|_{E_l}$ on boundary element $E^{\text {FS}}_l$. To evaluate $E_{\text {FS}}$ in Table 3, we need to evaluate the submatrix on boundary element $E^{\text {FS}}_l$ through

$$\begin{aligned} \int _{E^{{\text {FS}}}_l} {\overline{v}}^p_{h} \nu ^{f \rightarrow s}_j (u^s_{h})_j \,\mathrm {d}\Sigma = ({\tilde{v}}^p|_{E_l} )^{{\mathsf {H}}}[ J^{2D}_l \nu ^{f \rightarrow s}_j {\mathcal {M}}_{ps}^{2D} ] {\tilde{u}}^s_j|_{E_l} , \end{aligned}$$

(77)

upon extracting ${\tilde{v}}^p|_{E_l}$ on $E^{\text {FS}}_l$.

We are now able to build all the submatrices for the evaluation of the integrals in Table 3. We then assemble the global matrices from all these submatrices using standard techniques similar to those in [9, 67].

1.4 Construction of the Submatrices for the Perturbation of the Gravitational Potential

Similar to the previous subsections, we construct the submatrices in $C_s$ in Table 4,

$$\begin{aligned} \int _{K_k^{\text {S}}} \partial _{x_i} ( \rho ^0 (u^s_h)_i ) \,\mathrm {d}x&= ({\mathbf {1}}|_{K_k})^{{\mathsf {H}}}[J_k {\mathcal {M}}_s {\mathsf {D}}_i^s \rho ^0_k ] {\tilde{u}}^s_i|_{K_k} , \end{aligned}$$

(78)

$$\begin{aligned} \int _{E_l^{\text {FS}}} \nu ^{f\rightarrow s}_i (u^s_h)_i \left[ \rho ^0\right] ^s \,\mathrm {d}\Sigma&= ({\mathbf {1}}|_{E_l})^{{\mathsf {H}}}[ J^{2D}_l \nu ^{f\rightarrow s}_i [\rho ^0]^s_l {\mathcal {M}}_s^{2D} ] {\tilde{u}}^s_i|_{E_l} ,\end{aligned}$$

(79)

$$\begin{aligned} \int _{E_l^{\text {S}}} \nu _i (u^s_h)_i \left[ \rho ^0\right] ^+_- \,\mathrm {d}\Sigma&= ({\mathbf {1}}|_{E_l})^{{\mathsf {H}}}[J^{2D}_l \nu _i ([\rho ^0]^+_-)_l {\mathcal {M}}_s^{2D} ] {\tilde{u}}^s_i|_{E_l} , \end{aligned}$$

(80)

and the submatrices in $C_s^{{\mathsf {T}}}$,

$$\begin{aligned} \int _{K_k^{\text {S}}} [\partial _{x_i} ( \rho ^0 ({\overline{v}}^s_h)_i )] S_k(u_h) \,\mathrm {d}x&= ({\tilde{v}}^s_i|_{K_k})^{{\mathsf {H}}}[J_k \rho ^0_k ({\mathsf {D}}_i^s)^{{\mathsf {T}}}{\mathcal {M}}_s S_k({\tilde{u}}) ] {\mathbf {1}}|_{K_k}, \end{aligned}$$

(81)

$$\begin{aligned} \int _{E_l^{\text {FS}}} \nu ^{f\rightarrow s}_i ({\overline{v}}^s_h)_i S_l(u_h) \left[ \rho ^0\right] ^s \,\mathrm {d}\Sigma&= ({\tilde{v}}^s_i|_{E_l})^{{\mathsf {H}}}[ J^{2D}_l \nu ^{f\rightarrow s}_i {\mathcal {M}}_s^{2D} [\rho ^0]^s_l S_l({\tilde{u}}) ] {\mathbf {1}}|_{E_l} ,\end{aligned}$$

(82)

$$\begin{aligned} \int _{E_l^{\text {S}}} \nu _i ({\overline{v}}^s_h)_i S_l(u_h) \left[ \rho ^0\right] ^+_- \,\mathrm {d}\Sigma&= ({\tilde{v}}^s_i|_{E_l})^{{\mathsf {H}}}[J^{2D}_l \nu _i {\mathcal {M}}_s^{2D} ([\rho ^0]^+_-)_l S_l({\tilde{u}}) ] {\mathbf {1}}|_{E_l} , \end{aligned}$$

(83)

where ${\mathbf {1}}$ denotes a vector of all ones. The construction of the submatrices in $C_f$ and $C_f^{{\mathsf {T}}}$ is the same. We are now able to build all the submatrices for the evaluation of the integrals in Table 4.

Full Mode Coupling

Concerning the Galerkin approximation, we can use different, nonlocal bases of functions in the appropriate energy space, for example, the spectral-Galerkin method [123]. In this appendix, we consider the use of the eigenfunctions of a spherically symmetric, non-rotating, perfectly elastic and isotropic (SNREI) reference model as a basis in this method. This has been implemented by [39, 40, 142, 144], and named the full mode coupling approach. An immediate drawback of using this basis, however, is that the fluid–solid boundaries need to be spherically symmetric, as these are encoded in these basis functions.

We let $u_{km}$ represent the eigenfunctions associated with eigenfrequencies, $\omega _k$, in terms of spherical harmonics, $Y_l^m$, that is,

$$\begin{aligned} u_{km} = U_{km} {\mathbf {P}}_{lm} + V_{km} {\mathbf {B}}_{lm} + W_{km} {\mathbf {C}}_{lm} \quad \, \text {(no summation over }m), \end{aligned}$$

where k is the multi-index for the eigenfrequency; $m=-l,-l+1\ldots ,l-1,l$ is the index corresponding with the degeneracy with l denoting the spherical harmonic degree; $U_{km}, V_{km}$ and $W_{km}$ are the three components of eigenfunctions and are functions of the radial coordinate; ${\mathbf {P}}_{lm}$, ${\mathbf {B}}_{lm}$ and ${\mathbf {C}}_{lm}$ are the vector spherical harmonics, see [35, (8.36)] for their definition. In addition, $p_{km}$ needs to be introduced to constrain the solution, cf. (13) [37, Subsection 3.3]. Since $\nabla \cdot u_{km}(x)$ can be expanded using $Y_l^m(x)$ [35, (8.38)] and $u_{km}(x) \cdot g_{(r)}$ can also be expanded using $Y_l^m(x)$ for the radial models, we let $p_{km} = P_{km} Y_l^m$ with

$$\begin{aligned} P_{km} = - \kappa _{(r)} \left[ \partial _r U_{km} + r^{-1} (2 U_{km} - \sqrt{l(l+1)}V_{km}) \right] + \rho ^0_{(r)} g_{(r)} U_{km}, \end{aligned}$$

where $\rho _{(r)}^0$, $\kappa _{(r)}$ and $g_{(r)}$ denote the radial profiles of the density, bulk modulus and reference gravitational field of a radial model, respectively. Similarly, the incremental gravitational potential of the radial models takes the form, $s_{km} = S_{km} Y_l^m$, where $S_{km}$ is also a function in the radial coordinate. In the following, l and m are fixed.

Table 19 Implicit definition of the matrices in (87) (no summations over k and m). Since the construction of $A_{sg}^{(r)}$ is standard, we refer to [35, (8.43) & (8.44)] and [147, (3.1)]. In the above, $\int _{\Omega _{(r)}^{\text {S}}}=\sum _{q=1}^{N_L^{\text {S}}} \int _{L_q^{\text {S}}}$ and $\int _{\Omega _{(r)}^{\text {F}}}=\sum _{q=1}^{N_L^{\text {F}}} \int _{L_q^{\text {F}}}$

Table 20 Implicit definition of the matrices in (87) (no summation over k and m). In the above, $\int _{\Omega _{(r)}^{\text {S}}}=\sum _{q=1}^{N_L^{\text {S}}} \int _{L_q^{\text {S}}}$ and $\int _{\Omega _{(r)}^{\text {F}}}=\sum _{q=1}^{N_L^{\text {F}}} \int _{L_q^{\text {F}}}$. In the Poisson’s equation, the computation of the integral $\int _0^{\infty }$ requires special treatment, see [147, Chapter 3.2.2]

Table 21 Implicit definition of the matrices in (88) for the Cowling approximation

Table 22 Implicit definition of the matrices in (88)

In a SNREI model, for the computation of the toroidal modes, we only need to consider a solid annulus comprising the mantle and the crust. We exemplify the computations with the spheroidal modes and let $U'_{km}$, $P'_{km}$ and $S'_{km}$ be test functions for $U_{km}$, $P_{km}$ and $S_{km}$ following the Galerkin method. We let the ${\tilde{X}}_{(r)}$ be the 1D interval of the radial planet and have ${\tilde{X}}_{(r)} = \Omega _{(r)}^{\text {S}} \cup \Omega _{(r)}^{\text {F}}$, where $\Omega _{(r)}^{\text {S}}$ and $\Omega _{(r)}^{\text {F}}$ denote the 1D intervals for the solid and fluid regions, respectively. Given a regular finite-element partitioning ${\mathcal {T}}_h^{(r)}$ of the interval ${\tilde{X}}_{(r)}$, we denote an element of the mesh by $L_q \in {\mathcal {T}}_h^{(r)}$ and have ${\tilde{X}}_{(r)} = \bigcup _{q=1}^{N_L} L_q$, where $N_L$ denotes the total number of 1D elements. Furthermore, we let $L_q^{\text {S}}$ and $L_q^{\text {F}}$ specifically be elements in the solid and fluid regions and have

$$\begin{aligned} \Omega _{(r)}^{\text {S}} = \bigcup _{q=1}^{N_L^{\text {S}}} L_q^{\text {S}}, \quad \Omega _{(r)}^{\text {F}} = \bigcup _{q=1}^{N_L^{\text {F}}} L_q^{\text {F}}, \end{aligned}$$

where $N_L^{\text {S}}$ and $N_L^{\text {F}}$ denote the numbers of 1D elements in the solid and fluid regions, respectively. We let $\Sigma ^{\text {FS}}_{(r)}$ denote the fluid–solid boundary points in the radial interval. We introduce the finite-element solutions, $U_{km;h}^s$, $U_{km;h}^f$, $V_{km;h}^s$, $V_{km;h}^f$, $P_{km;h}$ and $S_{km;h}$, and test functions, $U_{km;h}^{s'}$, $U_{km;h}^{f'}$, $V_{km;h}^{s'}$, $V_{km;h}^{f'}$, $P'_{km;h}$ and $S'_{km;h}$. We set $N_{p^U} =(p^U+1)/2$, where $N_{p^U}$ is the number of nodes on a 1D element for the $p^U$-th order polynomial approximation. We have likewise expressions for $N_{p^V}$, $N_{p^P}$ and $N_{p^S}$. As in Sect. 4.2, we introduce nodal-based Lagrange polynomials, $\ell _i^{U}$, $\ell _i^{V}$, $\ell _i^{P}$, $\ell _i^S$, on the respective 1D elements $L \in {\mathcal {T}}_h^{(r)}$, and write

$$\begin{aligned} U_{km;h}^s(x)&= \sum _{i=1}^{N_{p^U}} U_{km;h}^s(x_i) \ell ^U_i(x) , \quad U_{km;h}^f(x) = \sum _{i=1}^{N_{p^U}} U_{km;h}^f(x_i) \ell ^U_i(x) , \end{aligned}$$

(84)

$$\begin{aligned} V_{km;h}^s(x)&= \sum _{i=1}^{N_{p^V}} V_{km;h}^s(x_i) \ell ^V_i(x) , \quad V_{km;h}^f(x) = \sum _{i=1}^{N_{p^V}} V_{km;h}^f(x_i) \ell ^V_i(x) , \end{aligned}$$

(85)

$$\begin{aligned} P_{km}(x)&= \sum _{i=1}^{N_{p^P}} P_{km}(x_i) \ell ^P_i(x), \quad S_{km}(x) = \sum _{i=1}^{N_{p^S}} S_{km}(x_i) \ell ^S_i(x), \end{aligned}$$

(86)

for $x\in L^{\text {S}}$ and $x \in L^{\text {F}}$, respectively; similar representations hold for $U_{km;h}^{s'}$, $U_{km;h}^{f'}$, $V_{km;h}^{s'}$, $V_{km;h}^{f'}$, $P'_{km;h}$ and $S'_{km;h}$, respectively. We note that the fluid–solid boundary points coincide with nodes.

As in Sects. 4 and 5, we collect the “values” of $U_{km;h}^s$, $U_{km;h}^f$, $V_{km;h}^s$, $V_{km;h}^f$, $P_{km;h}$ and $S_{km;h}$ at all the nodes, in vectors ${\tilde{U}}_{km}^s$, ${\tilde{U}}_{km}^f$, ${\tilde{V}}_{km}^s$, ${\tilde{V}}_{km}^f$, ${\tilde{V}}_{km}$ and ${\tilde{S}}_{km}$, respectively, and collect the values of $U_{km;h}^{s'}$, $U_{km;h}^{f'}$, $V_{km;h}^{s'}$, $V_{km;h}^{f'}$, $P'_{km;h}$ and $S'_{km;h}$ at all the nodes, in “vectors" ${\tilde{U}}_{km}^{s'}$, ${\tilde{U}}_{km}^{f'}$, ${\tilde{V}}_{km}^{s'}$, ${\tilde{V}}_{km}^{f'}$, ${\tilde{P}}'_{km}$ and ${\tilde{S}}'_{km}$, respectively. We let

$$\begin{aligned} {\tilde{u}}_{km}^{(r)}&= (({\tilde{U}}_{km}^s)^{{\mathsf {T}}}, ({\tilde{V}}_{km}^s)^{{\mathsf {T}}}, ({\tilde{U}}_{km}^f)^{{\mathsf {T}}}, ({\tilde{V}}_{km}^f)^{{\mathsf {T}}})^{{\mathsf {T}}}, \\ {\tilde{u}}^s_{km}&= (({\tilde{U}}^s_{km})^{{\mathsf {T}}}, ({\tilde{V}}^s_{km})^{{\mathsf {T}}})^{{\mathsf {T}}}, \quad {\tilde{u}}^f_{km} = (({\tilde{U}}^f_{km})^{{\mathsf {T}}}, ({\tilde{V}}^f_{km})^{{\mathsf {T}}})^{{\mathsf {T}}}, \end{aligned}$$

and obtain the resulting eigenvalue problem (cf. (54))

$$\begin{aligned} (A_{G}^{(r)} - E_{G}^{(r)} {A_p^{(r)}}^{-1} {E_{G}^{(r)}}^{{\mathsf {T}}}- {C^{(r)}}^{{\mathsf {T}}}(S^{(r)})^{-1} C^{(r)}) {\tilde{u}}_{km}^{(r)} = \omega _k^2 M^{(r)} {\tilde{u}}_{km}^{(r)} , \end{aligned}$$

(87)

where

$$\begin{aligned} A_{G}^{(r)} = \left( \begin{array}{cc} A_{sg}^{(r)} &{} 0 \\ 0 &{} A_f^{(r)} \end{array} \right) , \, E_G^{(r)}&= \left( \begin{array}{c} E_{\text {FS}}^{(r)} \\ A_{\text {dg}}^{(r)} \end{array} \right) , \, {C^{(r)}}^{{\mathsf {T}}}= \left( \begin{array}{c} {C_s^{(r)}}^{{\mathsf {T}}}\\ {C_f^{(r)}}^{{\mathsf {T}}}\end{array} \right) , \\ M^{(r)} = \left( \begin{array}{cc} M_s^{(r)} &{} 0 \\ 0 &{} M_f^{(r)} \end{array} \right) , \, {E_G^{(r)}}^{{\mathsf {T}}}&= \left( \begin{array}{cc} {E_{\text {FS}}^{(r)}}^{{\mathsf {T}}}&{A_{\text {dg}}^{(r)}}^{{\mathsf {T}}}\end{array} \right) , \, C^{(r)} = \left( \begin{array}{cc} C_s^{(r)}&C_f^{(r)} \end{array} \right) , \end{aligned}$$

in which $A_{sg}^{(r)}$, $A_f^{(r)}$, $A_p^{(r)}$, $E_{\text {FS}}^{(r)}$, ${E_{\text {FS}}^{(r)}}^{{\mathsf {T}}}$, $A_{\text {dg}}^{(r)}$, ${A_{\text {dg}}^{(r)}}^{{\mathsf {T}}}$, $M_s^{(r)}$, $M_f^{(r)}$, ${C_s^{(r)}}^{{\mathsf {T}}}$, ${C_f^{(r)}}^{{\mathsf {T}}}$, $S^{(r)}$, $C_s^{(r)}$ and $C_f^{(r)}$, are given in Tables 19 and 20. We note that the matrices in (87) are obtained using separation of variables with spherical harmonics in (54). We substitute

$$\begin{aligned} {\tilde{P}}_{km} =- {A_p^{(r)}}^{-1} {E_{G}^{(r)}}^{{\mathsf {T}}}{\tilde{u}}_{km}^{(r)} \end{aligned}$$

upon solving (17) and

$$\begin{aligned} {\tilde{S}}_{km} = (S^{(r)})^{-1} C^{(r)} {\tilde{u}}_{km}^{(r)} \end{aligned}$$

upon solving (2). We only need to invoke a finite-element basis in the radial coordinate. We note that the resulting system can be solved via a standard eigensolver, such as LAPACK [5].

As mentioned above, we may consider the finite-element solution denoted as $\{u_{km;h}\}$ as an alternative basis. Since $\{u_{km;h}\}$ is a global basis for the general problem, we have no separation in the solid and fluid components and no longer have the fluid–solid boundary terms in the system. Following the Galerkin method, we then consider an expansion for the general solution $u_c = \sum _{km} y_{km} u_{km;h}$ and the corresponding test functions $v_c = \sum _{k'm'} y'_{k'm'} u_{k'm';h}$. We introduce $s_c$ and its corresponding test functions $v^{s_c}$ for self-gravitation. We have $s_c = \sum _{km} z_{km} S_{km;h}$ and $v^{s_c} = \sum _{k'm'} z'_{k'm'} S_{k'm';h}$. Assuming that all the discontinuities in a fully heterogeneous model coincide with the ones in the reference radial model and the fluid outer core, the eigenfuncions represented by the mentioned expansions lie in $H_1 \subset E$ (cf. (32)) for the fully heterogeneous problem while the constraint equation disappears. We let y, $y'$, z and $z'$ be the “vectors" with components $y_{km}$, $y'_{k'm'}$, $z_{km}$ and $z'_{k'm'}$, respectively, and obtain

$$\begin{aligned} (A_{G}^{(c)} - {C^{(c)}}^{{\mathsf {T}}}{S^{(c)}}^{-1} C^{(c)}) y = \omega ^2 M^{(c)} y, \end{aligned}$$

(88)

as the counterpart of (54). Here, $A_{G}^{(c)}$, $M^{(c)}$, ${C^{(c)}}^{{\mathsf {T}}}$, $S^{(c)}$ and $C^{(c)}$, obtained via substituting the above-mentioned expansion of $u_c$ in (54), are given in Tables 21 and 22.

If all the discontinuities in a fully heterogeneous model with a fixed fluid outer core coincide with the reference radial model, we note that the matrix elements in (88), Tables 21 and 22 are similar to [142, (A1)], which describe mode coupling in non-radial models. However, Woodhouse [142, (A1)] includes additional terms accounting for changes in the fluid–solid boundaries while in the previous work [144, (42)], perturbation theory is used to compute the eigenfrequency changes in terms of the unperturbed eigenfunctions; both calculations violate the condition that normal modes need to remain in E and in $H_1$.