Gina Serpe Mon Feb 11, 3:25 AM ET

Los Angeles (E! Online) - The long and winding divorce of Paul McCartney and Heather Mills McCartney may finally be coming to an end.

The high-profile duo began the first day of what's expected to be a weeklong proceeding to hash out terms for their long-pending divorce settlement, a payout from McCartney to Mills McCartney that could be the largest in U.K. legal history.

The 40-year-old anti-mine campaigner was the first to arrive to the hearing Monday at the Family Division of London's High Court. She wore a grey and black pinstripe skirt suit and a pink blouse, and refused to speak to reporters on her way in. The court warded off the more brazen members of the press corps with a sign warning "No Admittance—Strictly Private." The windows of the courthouse were also blacked out.

The 65-year-old former Beatle arrived shortly after his soon-to-be ex-wife, also donning a grey pinstripe suit, bidding a quick "good morning" to the news media, though refusing to comment on the proceedings.

McCartney was accompanied by his attorney, Fiona Shackleton, who famously represented Prince Charles in his divorce from Princess Diana; Mills McCartney was solo.

The onetime model announced she would be representing herself in the case after firing her legal team (who, coincidentally had repped Diana her aforementioned split) last November. The termination came after several PR-misfire interviews on both British and U.S. television in which she railed against the alleged indifference of McCartney to her constant stream of death threats and thoughts of suicide, saying he failed both her and their daughter.

During this week's faceoff, Mills McCartney would be able to cross-examine her ex in front of the courld, should he take to the stand, just one of the perks of representing herself.

However, as the entire session is taking place in closed court, it's unlikely exact terms of any would-be settlement will ever be made public. The only way for the payout to become public knowledge would be if either McCartney or Mills McCartney blab to the press themselves (unlikely in the advent of confidentiality clauses) or if one of them contests the judgment in a court of appeals, in which case all proceedings would be made open to the public.

Any financial settlement itself will be rendered by High Court Judge Hugh Bennett, who will likely hold off on making a final judgment Friday and instead issue his ruling in a written statement.

Both sides have kept mum on any monetary offers in the two years since they separated, save for Mills McCartney's adamant denial of reports that she's holding out for more money. She's expected to rake in a minimum of $50 million in the deal—not bad for just four years of marriage. The couple share custody of four-year-old daughter Beatrice.

More generous estimates have claimed Mills McCartney may even see a payout of upwards of $100 million.

Although payout seems like massive bank, it would barely dent McCartney's estimated fortune of $1.6 billion.

In May 2006, the former twosome announced that they were amicably ending their four-year marriage. It didn't take long, however, before all semblance of amicability fell by the wayside, replaced by exceeding amounts of acrimony that played out in the headlines. The negative campaigning came to a head last October, when Mills McCartney revealed on Britain's GMTV that she had received "worse press than a pedophile or a murderer" and had at one point contemplated thoughts of suicide.

Her media manager resigned in the wake of the ill-advised press tour, and Mills McCartney was forced to fire her legal team, claiming she had gone into debt and could no longer afford to shell out for the attorneys.

Since the split, neither party has formally stepped out with a new partner, though McCartney has been linked with a string of women, most recently the actress Rosanna Arquette. (Both McCartney and Arquette said they were just friends.)

Meanwhile, McCartney's morning court date explains his absence from Sunday's Los Angeles-held Grammy Awards. The 50th anniversary ceremony featured a major tribute to the Fab Four, courtesy of performances from Love, the Cirque de Soleil show based on the Beatles' canon, and Across the Universe, the Julie Taymor film set to reworked Beatles' tunes.

In his stead, fellow former Beatle Ringo Starr pardoned his bandmate's absence and accepted the Grammy for Best Compilation Soundtrack for the Love on behalf of McCartney, himself and the two late Beatles, George Harrison and John Lennon.